| L(s) = 1 | − 5-s − 5·7-s − 5·13-s + 6·17-s − 5·19-s + 23-s − 4·25-s − 6·29-s + 10·31-s + 5·35-s − 4·43-s − 8·47-s + 18·49-s + 53-s − 3·59-s + 14·61-s + 5·65-s − 5·67-s + 9·71-s + 5·73-s + 79-s − 4·83-s − 6·85-s − 2·89-s + 25·91-s + 5·95-s + 8·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s − 1.38·13-s + 1.45·17-s − 1.14·19-s + 0.208·23-s − 4/5·25-s − 1.11·29-s + 1.79·31-s + 0.845·35-s − 0.609·43-s − 1.16·47-s + 18/7·49-s + 0.137·53-s − 0.390·59-s + 1.79·61-s + 0.620·65-s − 0.610·67-s + 1.06·71-s + 0.585·73-s + 0.112·79-s − 0.439·83-s − 0.650·85-s − 0.211·89-s + 2.62·91-s + 0.512·95-s + 0.812·97-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.17743684834807, −12.84306992693574, −12.29244311017231, −12.03751420899665, −11.62800560402360, −10.87469250250018, −10.28338819183157, −9.950904091598326, −9.616599843335984, −9.273953854362594, −8.493001070766395, −7.949444362272346, −7.665397636723883, −6.815487419292476, −6.740410647733494, −6.138072844489781, −5.493383170968289, −5.116355075778717, −4.293732016736449, −3.825430245309851, −3.387737366147656, −2.717357034528428, −2.422964249668047, −1.459520843650894, −0.5266121277975379, 0,
0.5266121277975379, 1.459520843650894, 2.422964249668047, 2.717357034528428, 3.387737366147656, 3.825430245309851, 4.293732016736449, 5.116355075778717, 5.493383170968289, 6.138072844489781, 6.740410647733494, 6.815487419292476, 7.665397636723883, 7.949444362272346, 8.493001070766395, 9.273953854362594, 9.616599843335984, 9.950904091598326, 10.28338819183157, 10.87469250250018, 11.62800560402360, 12.03751420899665, 12.29244311017231, 12.84306992693574, 13.17743684834807
