| L(s) = 1 | + 2·2-s + 4·3-s − 2·5-s + 8·6-s − 3·8-s − 9-s − 4·10-s + 8·11-s − 8·15-s − 6·16-s + 23·17-s − 2·18-s + 13·19-s + 16·22-s − 18·23-s − 12·24-s − 18·25-s − 26·27-s − 15·29-s − 16·30-s − 3·31-s − 2·32-s + 32·33-s + 46·34-s − 13·37-s + 26·38-s + 6·40-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s − 0.894·5-s + 3.26·6-s − 1.06·8-s − 1/3·9-s − 1.26·10-s + 2.41·11-s − 2.06·15-s − 3/2·16-s + 5.57·17-s − 0.471·18-s + 2.98·19-s + 3.41·22-s − 3.75·23-s − 2.44·24-s − 3.59·25-s − 5.00·27-s − 2.78·29-s − 2.92·30-s − 0.538·31-s − 0.353·32-s + 5.57·33-s + 7.88·34-s − 2.13·37-s + 4.21·38-s + 0.948·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.328320643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.328320643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p^{2} T^{2} - 5 T^{3} + 5 p T^{4} - 9 p T^{5} + 31 T^{6} - 9 p^{2} T^{7} + 5 p^{3} T^{8} - 5 p^{3} T^{9} + p^{6} T^{10} - p^{6} T^{11} + p^{6} T^{12} \) |
| 3 | \( 1 - 4 T + 17 T^{2} - 46 T^{3} + 122 T^{4} - 245 T^{5} + 481 T^{6} - 245 p T^{7} + 122 p^{2} T^{8} - 46 p^{3} T^{9} + 17 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 5 | \( 1 + 2 T + 22 T^{2} + 7 p T^{3} + 229 T^{4} + 303 T^{5} + 1447 T^{6} + 303 p T^{7} + 229 p^{2} T^{8} + 7 p^{4} T^{9} + 22 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 8 T + 50 T^{2} - 243 T^{3} + 1083 T^{4} - 4383 T^{5} + 15277 T^{6} - 4383 p T^{7} + 1083 p^{2} T^{8} - 243 p^{3} T^{9} + 50 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 23 T + 292 T^{2} - 2540 T^{3} + 16936 T^{4} - 91115 T^{5} + 409523 T^{6} - 91115 p T^{7} + 16936 p^{2} T^{8} - 2540 p^{3} T^{9} + 292 p^{4} T^{10} - 23 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 13 T + 161 T^{2} - 1215 T^{3} + 8665 T^{4} - 45671 T^{5} + 227327 T^{6} - 45671 p T^{7} + 8665 p^{2} T^{8} - 1215 p^{3} T^{9} + 161 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 18 T + 235 T^{2} + 2091 T^{3} + 15661 T^{4} + 94020 T^{5} + 495523 T^{6} + 94020 p T^{7} + 15661 p^{2} T^{8} + 2091 p^{3} T^{9} + 235 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 15 T + 183 T^{2} + 1577 T^{3} + 12663 T^{4} + 80831 T^{5} + 478995 T^{6} + 80831 p T^{7} + 12663 p^{2} T^{8} + 1577 p^{3} T^{9} + 183 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 3 T + 29 T^{2} - 99 T^{3} + 1365 T^{4} + 1247 T^{5} + 67011 T^{6} + 1247 p T^{7} + 1365 p^{2} T^{8} - 99 p^{3} T^{9} + 29 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 13 T + 227 T^{2} + 2287 T^{3} + 21261 T^{4} + 165075 T^{5} + 1053087 T^{6} + 165075 p T^{7} + 21261 p^{2} T^{8} + 2287 p^{3} T^{9} + 227 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 4 T + 131 T^{2} - 292 T^{3} + 6158 T^{4} - 3460 T^{5} + 202879 T^{6} - 3460 p T^{7} + 6158 p^{2} T^{8} - 292 p^{3} T^{9} + 131 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 18 T + 311 T^{2} + 3496 T^{3} + 35744 T^{4} + 284555 T^{5} + 2083101 T^{6} + 284555 p T^{7} + 35744 p^{2} T^{8} + 3496 p^{3} T^{9} + 311 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 16 T + 260 T^{2} - 2471 T^{3} + 24701 T^{4} - 179041 T^{5} + 1411093 T^{6} - 179041 p T^{7} + 24701 p^{2} T^{8} - 2471 p^{3} T^{9} + 260 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 25 T + 429 T^{2} + 5407 T^{3} + 58624 T^{4} + 523460 T^{5} + 4125969 T^{6} + 523460 p T^{7} + 58624 p^{2} T^{8} + 5407 p^{3} T^{9} + 429 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 18 T + 231 T^{2} + 2285 T^{3} + 20723 T^{4} + 164956 T^{5} + 1340761 T^{6} + 164956 p T^{7} + 20723 p^{2} T^{8} + 2285 p^{3} T^{9} + 231 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 16 T + 311 T^{2} + 3792 T^{3} + 44666 T^{4} + 408293 T^{5} + 3575773 T^{6} + 408293 p T^{7} + 44666 p^{2} T^{8} + 3792 p^{3} T^{9} + 311 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 16 T + 296 T^{2} - 2748 T^{3} + 28371 T^{4} - 190491 T^{5} + 1772499 T^{6} - 190491 p T^{7} + 28371 p^{2} T^{8} - 2748 p^{3} T^{9} + 296 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 25 T + 450 T^{2} - 5633 T^{3} + 66618 T^{4} - 665639 T^{5} + 6202577 T^{6} - 665639 p T^{7} + 66618 p^{2} T^{8} - 5633 p^{3} T^{9} + 450 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 5 T + 262 T^{2} - 1812 T^{3} + 33912 T^{4} - 262035 T^{5} + 2891423 T^{6} - 262035 p T^{7} + 33912 p^{2} T^{8} - 1812 p^{3} T^{9} + 262 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 2 T + 307 T^{2} - 689 T^{3} + 48091 T^{4} - 95350 T^{5} + 4742205 T^{6} - 95350 p T^{7} + 48091 p^{2} T^{8} - 689 p^{3} T^{9} + 307 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 7 T + 226 T^{2} - 2485 T^{3} + 35590 T^{4} - 306775 T^{5} + 3896483 T^{6} - 306775 p T^{7} + 35590 p^{2} T^{8} - 2485 p^{3} T^{9} + 226 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 10 T + 292 T^{2} - 2764 T^{3} + 48371 T^{4} - 414955 T^{5} + 5171855 T^{6} - 414955 p T^{7} + 48371 p^{2} T^{8} - 2764 p^{3} T^{9} + 292 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 5 T + 286 T^{2} - 1431 T^{3} + 39905 T^{4} - 196126 T^{5} + 4141037 T^{6} - 196126 p T^{7} + 39905 p^{2} T^{8} - 1431 p^{3} T^{9} + 286 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.93478185712026024907353293540, −3.70958975158847048807368818453, −3.56190686004302256150585509953, −3.49352698582333177741480923694, −3.44658586195466775053877831287, −3.39340887563787653719042899177, −3.38995077043887324241224755596, −3.13619189449916898305613519391, −3.13040643469762645845001388312, −2.91376347557383017175725771948, −2.78975755643585229072376101867, −2.53333751227807673595164870822, −2.24156567565112777678192872192, −2.21292605618553607767642079054, −2.14881630968431183844296440046, −1.84627864137763803750943074154, −1.64524046358266870718108333011, −1.61522951709809874709538010177, −1.56311805272832693358491782818, −1.32355642949915162275715396609, −1.15633930293753274803482416028, −0.76304914562188599355079184370, −0.53846730139699807561593077620, −0.34568709172730021143940698950, −0.14890988954381220288956742539,
0.14890988954381220288956742539, 0.34568709172730021143940698950, 0.53846730139699807561593077620, 0.76304914562188599355079184370, 1.15633930293753274803482416028, 1.32355642949915162275715396609, 1.56311805272832693358491782818, 1.61522951709809874709538010177, 1.64524046358266870718108333011, 1.84627864137763803750943074154, 2.14881630968431183844296440046, 2.21292605618553607767642079054, 2.24156567565112777678192872192, 2.53333751227807673595164870822, 2.78975755643585229072376101867, 2.91376347557383017175725771948, 3.13040643469762645845001388312, 3.13619189449916898305613519391, 3.38995077043887324241224755596, 3.39340887563787653719042899177, 3.44658586195466775053877831287, 3.49352698582333177741480923694, 3.56190686004302256150585509953, 3.70958975158847048807368818453, 3.93478185712026024907353293540
Plot not available for L-functions of degree greater than 10.
