| L(s) = 1 | + 18·5-s − 98·7-s + 396·11-s − 350·13-s − 1.80e3·17-s + 3.26e3·19-s + 2.08e3·23-s − 4.58e3·25-s − 6.69e3·29-s + 20·31-s − 1.76e3·35-s + 6.23e3·37-s + 6.04e3·41-s + 3.02e3·43-s + 1.17e4·47-s + 7.20e3·49-s − 9.46e3·53-s + 7.12e3·55-s − 4.39e4·59-s − 6.47e4·61-s − 6.30e3·65-s − 2.47e4·67-s + 9.74e4·71-s + 1.74e4·73-s − 3.88e4·77-s − 5.12e4·79-s + 1.17e5·83-s + ⋯ |
| L(s) = 1 | + 0.321·5-s − 0.755·7-s + 0.986·11-s − 0.574·13-s − 1.51·17-s + 2.07·19-s + 0.823·23-s − 1.46·25-s − 1.47·29-s + 0.00373·31-s − 0.243·35-s + 0.748·37-s + 0.561·41-s + 0.249·43-s + 0.772·47-s + 3/7·49-s − 0.462·53-s + 0.317·55-s − 1.64·59-s − 2.22·61-s − 0.184·65-s − 0.674·67-s + 2.29·71-s + 0.383·73-s − 0.745·77-s − 0.924·79-s + 1.87·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 18 T + 4906 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 36 p T + 142198 T^{2} - 36 p^{6} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 350 T + 546978 T^{2} + 350 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1800 T + 3567406 T^{2} + 1800 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3266 T + 7614270 T^{2} - 3266 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2088 T + 9365230 T^{2} - 2088 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6696 T + 51326470 T^{2} + 6696 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 20 T + 53103102 T^{2} - 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6232 T + 144242070 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6048 T + 223864366 T^{2} - 6048 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3020 T - 30383466 T^{2} - 3020 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11700 T + 292735582 T^{2} - 11700 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9468 T + 858185230 T^{2} + 9468 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 43938 T + 1852599934 T^{2} + 43938 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 64754 T + 2408321418 T^{2} + 64754 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24784 T + 2799959190 T^{2} + 24784 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 97416 T + 5729557966 T^{2} - 97416 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 17452 T + 3828622374 T^{2} - 17452 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 51256 T + 3645565854 T^{2} + 51256 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 117558 T + 7798161502 T^{2} - 117558 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 84276 T + 5915697430 T^{2} + 84276 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20776 T + 16174049358 T^{2} - 20776 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.175150302164513123790949954894, −9.003377073118348349123027797609, −7.985131466685477446769408539871, −7.72944461053358297963549532269, −7.40707532720373618194046179732, −6.88820179990317646367207800470, −6.39775635509878644450975966825, −6.19934223827698064471699527285, −5.41920780568823168578406540529, −5.40854320970119463119340355959, −4.42809873364306343492942112843, −4.34257277677079490080708863135, −3.42034170797955326635739367227, −3.41154352230595264692636382229, −2.48943303374564776679849230322, −2.24013419347387974551785045284, −1.36311277120176753384325492991, −1.08838532640191393328595686667, 0, 0,
1.08838532640191393328595686667, 1.36311277120176753384325492991, 2.24013419347387974551785045284, 2.48943303374564776679849230322, 3.41154352230595264692636382229, 3.42034170797955326635739367227, 4.34257277677079490080708863135, 4.42809873364306343492942112843, 5.40854320970119463119340355959, 5.41920780568823168578406540529, 6.19934223827698064471699527285, 6.39775635509878644450975966825, 6.88820179990317646367207800470, 7.40707532720373618194046179732, 7.72944461053358297963549532269, 7.985131466685477446769408539871, 9.003377073118348349123027797609, 9.175150302164513123790949954894
