| L(s) = 1 | − 3.10e3·7-s − 1.59e3·13-s + 7.10e4·19-s + 9.64e4·25-s − 1.55e5·31-s + 4.74e5·37-s + 5.24e4·43-s + 5.58e6·49-s + 8.46e5·61-s − 3.90e6·67-s − 6.94e6·73-s − 9.35e6·79-s + 4.96e6·91-s − 1.66e7·97-s + 1.05e7·103-s − 3.98e7·109-s + 3.73e6·121-s + ⋯ |
| L(s) = 1 | − 3.42·7-s − 0.201·13-s + 2.37·19-s + 1.23·25-s − 0.938·31-s + 1.54·37-s + 0.100·43-s + 6.78·49-s + 0.477·61-s − 1.58·67-s − 2.08·73-s − 2.13·79-s + 0.690·91-s − 1.84·97-s + 0.950·103-s − 2.94·109-s + 0.191·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 19294 p T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 1553 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3735338 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 799 T + p^{7} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 686988466 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 35533 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4441696226 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 12357437098 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 77828 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 237383 T + p^{7} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 246594376882 T^{2} + p^{14} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 26200 T + p^{7} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 860671848046 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 672543838406 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4939061126518 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 423059 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 1950023 T + p^{7} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 15579694271662 T^{2} + p^{14} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3470239 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4678583 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 50603593187254 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 81637003580578 T^{2} + p^{14} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8300083 T + p^{7} T^{2} )^{2} \) |
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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
−9.665121763858064624487686325723, −9.455263480989169553704982399448, −9.030969074227942126776768953339, −8.684013532133976694275871622610, −7.56260707446392180650615810875, −7.49789175904293203919515145693, −6.84712795077255124762503275250, −6.61280223605730513742701124817, −5.99110175616991630321779071217, −5.67125135829249229961944844165, −5.16854162250047366691838263021, −4.23330876947410632022382461384, −3.79293358557300724386913718983, −3.16543940565260643951434219201, −2.83335225101374503878042140958, −2.70126408719179299716524898642, −1.33627441507231842213392483490, −0.898662765821912166188341588300, 0, 0,
0.898662765821912166188341588300, 1.33627441507231842213392483490, 2.70126408719179299716524898642, 2.83335225101374503878042140958, 3.16543940565260643951434219201, 3.79293358557300724386913718983, 4.23330876947410632022382461384, 5.16854162250047366691838263021, 5.67125135829249229961944844165, 5.99110175616991630321779071217, 6.61280223605730513742701124817, 6.84712795077255124762503275250, 7.49789175904293203919515145693, 7.56260707446392180650615810875, 8.684013532133976694275871622610, 9.030969074227942126776768953339, 9.455263480989169553704982399448, 9.665121763858064624487686325723
