| L(s) = 1 | − 5-s − 7-s + 11-s − 4·13-s − 6·17-s + 6·19-s − 8·23-s + 25-s + 35-s − 2·37-s + 8·41-s − 8·43-s + 49-s − 10·53-s − 55-s − 4·59-s + 2·61-s + 4·65-s − 12·67-s + 8·71-s − 12·73-s − 77-s − 10·79-s + 14·83-s + 6·85-s + 10·89-s + 4·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.301·11-s − 1.10·13-s − 1.45·17-s + 1.37·19-s − 1.66·23-s + 1/5·25-s + 0.169·35-s − 0.328·37-s + 1.24·41-s − 1.21·43-s + 1/7·49-s − 1.37·53-s − 0.134·55-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-s + 0.949·71-s − 1.40·73-s − 0.113·77-s − 1.12·79-s + 1.53·83-s + 0.650·85-s + 1.05·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.18581925111871, −12.62697050653945, −12.24262586717158, −11.82865180863130, −11.40189115823965, −10.99246313113322, −10.27943851246537, −9.899277688598478, −9.535497865821335, −8.941509946230043, −8.600169646157393, −7.797902719721022, −7.566601372354157, −7.111855986086512, −6.392785439981568, −6.182454129356906, −5.426143567504832, −4.864871386173391, −4.413224335155650, −3.937884004177497, −3.244031616049269, −2.822861724610721, −2.081838374022044, −1.611803318426621, −0.5921141944437097, 0,
0.5921141944437097, 1.611803318426621, 2.081838374022044, 2.822861724610721, 3.244031616049269, 3.937884004177497, 4.413224335155650, 4.864871386173391, 5.426143567504832, 6.182454129356906, 6.392785439981568, 7.111855986086512, 7.566601372354157, 7.797902719721022, 8.600169646157393, 8.941509946230043, 9.535497865821335, 9.899277688598478, 10.27943851246537, 10.99246313113322, 11.40189115823965, 11.82865180863130, 12.24262586717158, 12.62697050653945, 13.18581925111871
