| L(s) = 1 | + 3·5-s − 3·7-s + 6·17-s + 4·19-s + 23-s + 4·25-s − 2·29-s − 3·31-s − 9·35-s − 8·37-s + 5·41-s − 5·43-s + 2·49-s − 53-s + 10·59-s + 14·61-s + 6·67-s + 10·71-s − 4·73-s + 5·79-s − 8·83-s + 18·85-s − 3·89-s + 12·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s + 1.45·17-s + 0.917·19-s + 0.208·23-s + 4/5·25-s − 0.371·29-s − 0.538·31-s − 1.52·35-s − 1.31·37-s + 0.780·41-s − 0.762·43-s + 2/7·49-s − 0.137·53-s + 1.30·59-s + 1.79·61-s + 0.733·67-s + 1.18·71-s − 0.468·73-s + 0.562·79-s − 0.878·83-s + 1.95·85-s − 0.317·89-s + 1.23·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.277483444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.277483444\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.96408113961763, −12.75957408039007, −12.27866665160132, −11.65132563980690, −11.22657267111670, −10.51801488702314, −9.976515504790256, −9.872329949028514, −9.489652082241191, −8.908335280780193, −8.437713446890308, −7.720823109791255, −7.230465372834414, −6.725904640841982, −6.312076230820142, −5.683843485223392, −5.348514890160246, −5.078244830120756, −3.973701790158756, −3.529500176950543, −3.080522848862168, −2.452369837860178, −1.847896640047937, −1.192420481779398, −0.5317627419234379,
0.5317627419234379, 1.192420481779398, 1.847896640047937, 2.452369837860178, 3.080522848862168, 3.529500176950543, 3.973701790158756, 5.078244830120756, 5.348514890160246, 5.683843485223392, 6.312076230820142, 6.725904640841982, 7.230465372834414, 7.720823109791255, 8.437713446890308, 8.908335280780193, 9.489652082241191, 9.872329949028514, 9.976515504790256, 10.51801488702314, 11.22657267111670, 11.65132563980690, 12.27866665160132, 12.75957408039007, 12.96408113961763
