| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 14-s + 16-s − 2·17-s + 20-s − 6·23-s + 25-s + 28-s + 6·29-s + 2·31-s − 32-s + 2·34-s + 35-s + 2·37-s − 40-s − 10·41-s − 12·43-s + 6·46-s + 4·47-s + 49-s − 50-s + 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.328·37-s − 0.158·40-s − 1.56·41-s − 1.82·43-s + 0.884·46-s + 0.583·47-s + 1/7·49-s − 0.141·50-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.89416539439826, −13.47272138017906, −13.07259448236264, −12.25157417200776, −11.84277175722777, −11.65084486871963, −10.76425028504043, −10.51555304288078, −9.912878366811480, −9.675913641981502, −8.878935293837134, −8.505791519845330, −8.091679671964767, −7.584783981943678, −6.732512175069137, −6.595445085666447, −5.985577257759292, −5.145761055430341, −4.974129044078559, −3.999889326112166, −3.581221457814588, −2.629166122925249, −2.241092860916720, −1.581791023774416, −0.8831648898504399, 0,
0.8831648898504399, 1.581791023774416, 2.241092860916720, 2.629166122925249, 3.581221457814588, 3.999889326112166, 4.974129044078559, 5.145761055430341, 5.985577257759292, 6.595445085666447, 6.732512175069137, 7.584783981943678, 8.091679671964767, 8.505791519845330, 8.878935293837134, 9.675913641981502, 9.912878366811480, 10.51555304288078, 10.76425028504043, 11.65084486871963, 11.84277175722777, 12.25157417200776, 13.07259448236264, 13.47272138017906, 13.89416539439826
