| L(s) = 1 | + 3-s − 2·5-s − 2·7-s + 9-s + 2·11-s − 13-s − 2·15-s − 3·17-s − 2·21-s − 3·23-s − 25-s + 27-s − 2·29-s − 5·31-s + 2·33-s + 4·35-s − 37-s − 39-s − 9·41-s + 43-s − 2·45-s − 2·47-s − 3·49-s − 3·51-s − 6·53-s − 4·55-s − 11·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.516·15-s − 0.727·17-s − 0.436·21-s − 0.625·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.898·31-s + 0.348·33-s + 0.676·35-s − 0.164·37-s − 0.160·39-s − 1.40·41-s + 0.152·43-s − 0.298·45-s − 0.291·47-s − 3/7·49-s − 0.420·51-s − 0.824·53-s − 0.539·55-s − 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−14.03880871299434, −13.71997565679366, −13.09030563409982, −12.64929558249302, −12.29529828105606, −11.58607148095936, −11.39074873625962, −10.71687239354876, −10.07964752961619, −9.698463160352695, −9.150920456826768, −8.743653897926239, −8.165516366002780, −7.715320713059549, −7.134072371637887, −6.756682658772119, −6.174106224735929, −5.580185425476100, −4.768633371185101, −4.271734202512219, −3.756455872214674, −3.309345748760863, −2.746186549612492, −1.907952804424209, −1.389272371121301, 0, 0,
1.389272371121301, 1.907952804424209, 2.746186549612492, 3.309345748760863, 3.756455872214674, 4.271734202512219, 4.768633371185101, 5.580185425476100, 6.174106224735929, 6.756682658772119, 7.134072371637887, 7.715320713059549, 8.165516366002780, 8.743653897926239, 9.150920456826768, 9.698463160352695, 10.07964752961619, 10.71687239354876, 11.39074873625962, 11.58607148095936, 12.29529828105606, 12.64929558249302, 13.09030563409982, 13.71997565679366, 14.03880871299434
