| L(s) = 1 | + 12·3-s − 50·5-s + 52·7-s + 138·9-s − 560·11-s − 1.38e3·13-s − 600·15-s + 148·17-s + 1.00e3·19-s + 624·21-s − 2.45e3·23-s + 1.87e3·25-s + 4.50e3·27-s − 1.34e3·29-s − 2.24e3·31-s − 6.72e3·33-s − 2.60e3·35-s + 5.94e3·37-s − 1.66e4·39-s + 2.30e4·41-s − 1.76e4·43-s − 6.90e3·45-s − 2.90e3·47-s − 2.69e4·49-s + 1.77e3·51-s + 5.41e3·53-s + 2.80e4·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.894·5-s + 0.401·7-s + 0.567·9-s − 1.39·11-s − 2.27·13-s − 0.688·15-s + 0.124·17-s + 0.635·19-s + 0.308·21-s − 0.966·23-s + 3/5·25-s + 1.18·27-s − 0.295·29-s − 0.420·31-s − 1.07·33-s − 0.358·35-s + 0.713·37-s − 1.75·39-s + 2.14·41-s − 1.45·43-s − 0.507·45-s − 0.192·47-s − 1.60·49-s + 0.0956·51-s + 0.264·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 4 p T + 2 p T^{2} - 4 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 52 T + 29646 T^{2} - 52 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 560 T + 381926 T^{2} + 560 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1388 T + 1149918 T^{2} + 1388 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 148 T - 795706 T^{2} - 148 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1000 T + 4533462 T^{2} - 1000 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2452 T + 6569198 T^{2} + 2452 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1340 T - 8758306 T^{2} + 1340 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2248 T + 57017022 T^{2} + 2248 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5940 T + 123434318 T^{2} - 5940 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17684 T + 312898614 T^{2} + 17684 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2908 T + 56660030 T^{2} + 2908 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5412 T + 693247822 T^{2} - 5412 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 62584 T + 2277965606 T^{2} + 62584 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 14108 T + 1110042462 T^{2} + 14108 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 85412 T + 4371910566 T^{2} - 85412 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 47208 T + 4011779662 T^{2} - 47208 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 924 p T + 4400780438 T^{2} + 924 p^{6} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 65904 T + 3994274078 T^{2} + 65904 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 108724 T + 10572459494 T^{2} + 108724 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 55020 T + 10818978262 T^{2} + 55020 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 147668 T + 11612429670 T^{2} - 147668 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.45947058434478772022474981015, −10.01922427600089407112836379233, −9.657531629097332602211321910078, −9.282235898816320923547648625365, −8.456292417091123377035563213955, −8.103754572222881226220997678658, −7.60267796217372383669773747510, −7.55210304967867538391863570003, −6.96139724634121783584378548629, −6.20229177997926447256767719308, −5.33989874029378919615111521975, −4.98564328454176858249239791859, −4.49594351488638667210590265303, −3.94496176282658847171563747832, −2.96809213773202582632171074880, −2.78732980976682471581089184963, −2.10645138909285204255257630129, −1.23787085294034386920698253609, 0, 0,
1.23787085294034386920698253609, 2.10645138909285204255257630129, 2.78732980976682471581089184963, 2.96809213773202582632171074880, 3.94496176282658847171563747832, 4.49594351488638667210590265303, 4.98564328454176858249239791859, 5.33989874029378919615111521975, 6.20229177997926447256767719308, 6.96139724634121783584378548629, 7.55210304967867538391863570003, 7.60267796217372383669773747510, 8.103754572222881226220997678658, 8.456292417091123377035563213955, 9.282235898816320923547648625365, 9.657531629097332602211321910078, 10.01922427600089407112836379233, 10.45947058434478772022474981015
