| L(s) = 1 | − 2·2-s − 4·3-s − 6·5-s + 8·6-s + 7·7-s + 4·8-s + 4·9-s + 12·10-s + 15·11-s − 6·13-s − 14·14-s + 24·15-s − 3·16-s + 5·17-s − 8·18-s − 28·21-s − 30·22-s − 16·24-s + 21·25-s + 12·26-s + 20·29-s − 48·30-s − 12·31-s − 2·32-s − 60·33-s − 10·34-s − 42·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s − 2.68·5-s + 3.26·6-s + 2.64·7-s + 1.41·8-s + 4/3·9-s + 3.79·10-s + 4.52·11-s − 1.66·13-s − 3.74·14-s + 6.19·15-s − 3/4·16-s + 1.21·17-s − 1.88·18-s − 6.11·21-s − 6.39·22-s − 3.26·24-s + 21/5·25-s + 2.35·26-s + 3.71·29-s − 8.76·30-s − 2.15·31-s − 0.353·32-s − 10.4·33-s − 1.71·34-s − 7.09·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.177927580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177927580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 + T )^{6} \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T + p^{2} T^{2} + p^{2} T^{3} + 3 T^{4} - p T^{5} - p^{3} T^{6} - p^{2} T^{7} + 3 p^{2} T^{8} + p^{5} T^{9} + p^{6} T^{10} + p^{6} T^{11} + p^{6} T^{12} \) |
| 3 | \( 1 + 4 T + 4 p T^{2} + 32 T^{3} + 67 T^{4} + 128 T^{5} + 241 T^{6} + 128 p T^{7} + 67 p^{2} T^{8} + 32 p^{3} T^{9} + 4 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - p T + 40 T^{2} - 170 T^{3} + 663 T^{4} - 2088 T^{5} + 5989 T^{6} - 2088 p T^{7} + 663 p^{2} T^{8} - 170 p^{3} T^{9} + 40 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 15 T + 119 T^{2} - 589 T^{3} + 2003 T^{4} - 5018 T^{5} + 13434 T^{6} - 5018 p T^{7} + 2003 p^{2} T^{8} - 589 p^{3} T^{9} + 119 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 6 T + 59 T^{2} + 210 T^{3} + 1275 T^{4} + 3264 T^{5} + 17906 T^{6} + 3264 p T^{7} + 1275 p^{2} T^{8} + 210 p^{3} T^{9} + 59 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 5 T + 63 T^{2} - 309 T^{3} + 2087 T^{4} - 8966 T^{5} + 44354 T^{6} - 8966 p T^{7} + 2087 p^{2} T^{8} - 309 p^{3} T^{9} + 63 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 70 T^{2} + 152 T^{3} + 1665 T^{4} + 10360 T^{5} + 26925 T^{6} + 10360 p T^{7} + 1665 p^{2} T^{8} + 152 p^{3} T^{9} + 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( 1 - 20 T + 245 T^{2} - 2024 T^{3} + 13140 T^{4} - 71610 T^{5} + 381572 T^{6} - 71610 p T^{7} + 13140 p^{2} T^{8} - 2024 p^{3} T^{9} + 245 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 12 T + 197 T^{2} + 1684 T^{3} + 15783 T^{4} + 99352 T^{5} + 659478 T^{6} + 99352 p T^{7} + 15783 p^{2} T^{8} + 1684 p^{3} T^{9} + 197 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 20 T + 369 T^{2} + 4188 T^{3} + 42967 T^{4} + 328096 T^{5} + 2270238 T^{6} + 328096 p T^{7} + 42967 p^{2} T^{8} + 4188 p^{3} T^{9} + 369 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 + 8 T + 234 T^{2} + 1462 T^{3} + 23099 T^{4} + 113050 T^{5} + 1245401 T^{6} + 113050 p T^{7} + 23099 p^{2} T^{8} + 1462 p^{3} T^{9} + 234 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 27 T + 531 T^{2} - 7105 T^{3} + 77655 T^{4} - 669698 T^{5} + 4874202 T^{6} - 669698 p T^{7} + 77655 p^{2} T^{8} - 7105 p^{3} T^{9} + 531 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 + 8 T + 206 T^{2} + 1416 T^{3} + 18289 T^{4} + 111272 T^{5} + 1016733 T^{6} + 111272 p T^{7} + 18289 p^{2} T^{8} + 1416 p^{3} T^{9} + 206 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 19 T + 415 T^{2} + 4919 T^{3} + 60491 T^{4} + 509538 T^{5} + 4373786 T^{6} + 509538 p T^{7} + 60491 p^{2} T^{8} + 4919 p^{3} T^{9} + 415 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 41 T + 993 T^{2} - 16763 T^{3} + 218583 T^{4} - 2272006 T^{5} + 19296702 T^{6} - 2272006 p T^{7} + 218583 p^{2} T^{8} - 16763 p^{3} T^{9} + 993 p^{4} T^{10} - 41 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 6 T + 185 T^{2} - 644 T^{3} + 19388 T^{4} - 60982 T^{5} + 1454112 T^{6} - 60982 p T^{7} + 19388 p^{2} T^{8} - 644 p^{3} T^{9} + 185 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 15 T + 388 T^{2} - 4384 T^{3} + 63727 T^{4} - 550286 T^{5} + 5656519 T^{6} - 550286 p T^{7} + 63727 p^{2} T^{8} - 4384 p^{3} T^{9} + 388 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 + 2 T + 143 T^{2} - 126 T^{3} + 16135 T^{4} + 416 T^{5} + 1451042 T^{6} + 416 p T^{7} + 16135 p^{2} T^{8} - 126 p^{3} T^{9} + 143 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 8 T + 369 T^{2} - 2348 T^{3} + 59119 T^{4} - 301932 T^{5} + 5477102 T^{6} - 301932 p T^{7} + 59119 p^{2} T^{8} - 2348 p^{3} T^{9} + 369 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 18 T + 483 T^{2} + 6226 T^{3} + 95411 T^{4} + 925328 T^{5} + 10032274 T^{6} + 925328 p T^{7} + 95411 p^{2} T^{8} + 6226 p^{3} T^{9} + 483 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 + 10 T + 219 T^{2} + 1886 T^{3} + 28951 T^{4} + 215212 T^{5} + 2956362 T^{6} + 215212 p T^{7} + 28951 p^{2} T^{8} + 1886 p^{3} T^{9} + 219 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 + 17 T + 532 T^{2} + 6690 T^{3} + 117497 T^{4} + 1122958 T^{5} + 13892321 T^{6} + 1122958 p T^{7} + 117497 p^{2} T^{8} + 6690 p^{3} T^{9} + 532 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 4 T + 277 T^{2} - 1600 T^{3} + 40415 T^{4} - 293644 T^{5} + 4254566 T^{6} - 293644 p T^{7} + 40415 p^{2} T^{8} - 1600 p^{3} T^{9} + 277 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−4.88023205234692816408586259284, −4.59394483942419866555739683936, −4.52554252470992054593629553925, −4.34900122391847587230392065927, −4.25896102020586437808993893389, −4.10061747756962679420732962148, −4.08049587743345078750588549359, −4.04075382327295068699718893255, −3.57766416898384045398258269275, −3.41677591342421968464749603418, −3.41216879442282087293091122533, −3.34397026611186199247240705411, −2.82780512370617744140575282410, −2.81021154837284290174481210990, −2.68246643056519717433082921685, −1.89226926110354005974963300248, −1.83742267367235671262150023896, −1.81474848050861129905345706513, −1.72356786553211435473065316434, −1.48156027491688597429237378835, −1.05485392267289955809435740553, −0.815632238808552333855961062501, −0.56889876388102583060481163389, −0.55008831095938739088069262877, −0.52676908828243685798429048782,
0.52676908828243685798429048782, 0.55008831095938739088069262877, 0.56889876388102583060481163389, 0.815632238808552333855961062501, 1.05485392267289955809435740553, 1.48156027491688597429237378835, 1.72356786553211435473065316434, 1.81474848050861129905345706513, 1.83742267367235671262150023896, 1.89226926110354005974963300248, 2.68246643056519717433082921685, 2.81021154837284290174481210990, 2.82780512370617744140575282410, 3.34397026611186199247240705411, 3.41216879442282087293091122533, 3.41677591342421968464749603418, 3.57766416898384045398258269275, 4.04075382327295068699718893255, 4.08049587743345078750588549359, 4.10061747756962679420732962148, 4.25896102020586437808993893389, 4.34900122391847587230392065927, 4.52554252470992054593629553925, 4.59394483942419866555739683936, 4.88023205234692816408586259284
Plot not available for L-functions of degree greater than 10.
