| L(s) = 1 | + 2·3-s − 2·4-s − 3·5-s + 7-s + 9-s − 3·11-s − 4·12-s − 6·15-s + 4·16-s − 3·17-s + 6·20-s + 2·21-s + 4·25-s − 4·27-s − 2·28-s − 6·29-s − 4·31-s − 6·33-s − 3·35-s − 2·36-s + 2·37-s − 6·41-s − 43-s + 6·44-s − 3·45-s + 3·47-s + 8·48-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.15·12-s − 1.54·15-s + 16-s − 0.727·17-s + 1.34·20-s + 0.436·21-s + 4/5·25-s − 0.769·27-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 1.04·33-s − 0.507·35-s − 1/3·36-s + 0.328·37-s − 0.937·41-s − 0.152·43-s + 0.904·44-s − 0.447·45-s + 0.437·47-s + 1.15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.88369593851261, −14.24415425281834, −13.90644133122915, −13.25530990670568, −12.92062081213827, −12.49228243861673, −11.70918834513442, −11.26015748191250, −10.79522652596576, −10.11041475120276, −9.467705741268656, −9.025201115403672, −8.579423899300211, −8.057226663331981, −7.729148982923338, −7.421358536641740, −6.505845089631629, −5.667717850256095, −5.020337886428173, −4.573919202230916, −3.911202430945555, −3.512797931090077, −2.972028033133363, −2.180114389857824, −1.385289990851800, 0, 0,
1.385289990851800, 2.180114389857824, 2.972028033133363, 3.512797931090077, 3.911202430945555, 4.573919202230916, 5.020337886428173, 5.667717850256095, 6.505845089631629, 7.421358536641740, 7.729148982923338, 8.057226663331981, 8.579423899300211, 9.025201115403672, 9.467705741268656, 10.11041475120276, 10.79522652596576, 11.26015748191250, 11.70918834513442, 12.49228243861673, 12.92062081213827, 13.25530990670568, 13.90644133122915, 14.24415425281834, 14.88369593851261
