| L(s) = 1 | + 5-s − 5·11-s + 13-s − 3·17-s + 3·23-s + 25-s + 5·29-s + 31-s − 2·37-s − 6·41-s − 43-s + 9·47-s − 7·49-s + 2·53-s − 5·55-s + 12·59-s − 6·61-s + 65-s + 8·67-s + 8·71-s − 16·73-s − 79-s + 2·83-s − 3·85-s − 4·89-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.50·11-s + 0.277·13-s − 0.727·17-s + 0.625·23-s + 1/5·25-s + 0.928·29-s + 0.179·31-s − 0.328·37-s − 0.937·41-s − 0.152·43-s + 1.31·47-s − 49-s + 0.274·53-s − 0.674·55-s + 1.56·59-s − 0.768·61-s + 0.124·65-s + 0.977·67-s + 0.949·71-s − 1.87·73-s − 0.112·79-s + 0.219·83-s − 0.325·85-s − 0.423·89-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−16.00614025372550, −15.66271015355450, −15.10342088934757, −14.51539335077359, −13.68126245148624, −13.50335269726635, −12.88117290074863, −12.39722660006501, −11.62850514532601, −11.02575285585950, −10.48171843823121, −10.08246797216746, −9.400689400313364, −8.587955028959563, −8.348054394271076, −7.492614079766164, −6.906073440697384, −6.292753053812917, −5.520382419618456, −5.045374448947065, −4.408114113758932, −3.450969754434169, −2.701963538537407, −2.162519823300761, −1.109583030209035, 0,
1.109583030209035, 2.162519823300761, 2.701963538537407, 3.450969754434169, 4.408114113758932, 5.045374448947065, 5.520382419618456, 6.292753053812917, 6.906073440697384, 7.492614079766164, 8.348054394271076, 8.587955028959563, 9.400689400313364, 10.08246797216746, 10.48171843823121, 11.02575285585950, 11.62850514532601, 12.39722660006501, 12.88117290074863, 13.50335269726635, 13.68126245148624, 14.51539335077359, 15.10342088934757, 15.66271015355450, 16.00614025372550
