| L(s) = 1 | − 4·2-s + 12·4-s − 32·8-s + 34·9-s + 40·11-s + 80·16-s − 136·18-s − 160·22-s − 96·23-s − 332·29-s − 192·32-s + 408·36-s + 156·37-s − 872·43-s + 480·44-s + 384·46-s − 124·53-s + 1.32e3·58-s + 448·64-s − 1.16e3·67-s − 1.08e3·71-s − 1.08e3·72-s − 624·74-s − 1.36e3·79-s + 427·81-s + 3.48e3·86-s − 1.28e3·88-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1.25·9-s + 1.09·11-s + 5/4·16-s − 1.78·18-s − 1.55·22-s − 0.870·23-s − 2.12·29-s − 1.06·32-s + 17/9·36-s + 0.693·37-s − 3.09·43-s + 1.64·44-s + 1.23·46-s − 0.321·53-s + 3.00·58-s + 7/8·64-s − 2.11·67-s − 1.81·71-s − 1.78·72-s − 0.980·74-s − 1.93·79-s + 0.585·81-s + 4.37·86-s − 1.55·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 34 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 82 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6658 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 13630 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 16990 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 17390 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 436 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 165054 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 32850 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 379954 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 580 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 544 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 417586 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 680 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1104766 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 842862 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1394146 T^{2} + p^{6} T^{4} \) |
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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
−8.356521496347212196363996474958, −8.203749629201019659314624594035, −7.38210926602821896058941986305, −7.32298632832326654596333515262, −7.28489672987726906126024162258, −6.51433453081464948754537684446, −6.15248300663623477870565263125, −6.02873965359463346612004312342, −5.34190977259151554648769470512, −4.79618027490603302055687902302, −4.31415500592024781148753263975, −3.92586778814333215034839788131, −3.40638401145371482322114357783, −3.04961695700585160522120906798, −2.22006355507909600984006538865, −1.66659494523455859441770297216, −1.58269001665792977889560344222, −1.02649757309148569338538015999, 0, 0,
1.02649757309148569338538015999, 1.58269001665792977889560344222, 1.66659494523455859441770297216, 2.22006355507909600984006538865, 3.04961695700585160522120906798, 3.40638401145371482322114357783, 3.92586778814333215034839788131, 4.31415500592024781148753263975, 4.79618027490603302055687902302, 5.34190977259151554648769470512, 6.02873965359463346612004312342, 6.15248300663623477870565263125, 6.51433453081464948754537684446, 7.28489672987726906126024162258, 7.32298632832326654596333515262, 7.38210926602821896058941986305, 8.203749629201019659314624594035, 8.356521496347212196363996474958
