| L(s) = 1 | + 5-s + 11-s − 13-s − 4·17-s + 2·19-s − 4·23-s + 25-s + 6·29-s − 8·37-s + 6·41-s + 4·43-s − 7·49-s + 6·53-s + 55-s − 12·59-s + 6·61-s − 65-s + 10·67-s − 12·71-s + 2·73-s + 8·79-s + 4·83-s − 4·85-s − 6·89-s + 2·95-s − 16·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.277·13-s − 0.970·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.31·37-s + 0.937·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.134·55-s − 1.56·59-s + 0.768·61-s − 0.124·65-s + 1.22·67-s − 1.42·71-s + 0.234·73-s + 0.900·79-s + 0.439·83-s − 0.433·85-s − 0.635·89-s + 0.205·95-s − 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−14.03474040091694, −13.58386421987227, −12.99968572484375, −12.42144690474116, −12.16161054889777, −11.45600106628998, −11.10804985805664, −10.42670330428423, −10.08949141672766, −9.524663654966544, −9.031974491539825, −8.588229470478498, −8.019942908395521, −7.409926666590762, −6.902231878568163, −6.335878277332688, −5.972328738980508, −5.225275504839093, −4.776304080932998, −4.171577014768518, −3.586117730736944, −2.841594488498453, −2.290414381733353, −1.676634040381187, −0.9066928021582388, 0,
0.9066928021582388, 1.676634040381187, 2.290414381733353, 2.841594488498453, 3.586117730736944, 4.171577014768518, 4.776304080932998, 5.225275504839093, 5.972328738980508, 6.335878277332688, 6.902231878568163, 7.409926666590762, 8.019942908395521, 8.588229470478498, 9.031974491539825, 9.524663654966544, 10.08949141672766, 10.42670330428423, 11.10804985805664, 11.45600106628998, 12.16161054889777, 12.42144690474116, 12.99968572484375, 13.58386421987227, 14.03474040091694
