| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 3·7-s − 3·8-s + 9-s − 5·11-s + 12-s + 3·14-s − 16-s + 2·17-s + 18-s − 6·19-s − 3·21-s − 5·22-s + 2·23-s + 3·24-s − 27-s − 3·28-s − 8·29-s − 10·31-s + 5·32-s + 5·33-s + 2·34-s − 36-s − 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.13·7-s − 1.06·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s + 0.801·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.37·19-s − 0.654·21-s − 1.06·22-s + 0.417·23-s + 0.612·24-s − 0.192·27-s − 0.566·28-s − 1.48·29-s − 1.79·31-s + 0.883·32-s + 0.870·33-s + 0.342·34-s − 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 43 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.91904082399060, −13.25187152959706, −12.87977629194305, −12.63299895326357, −12.21163684558233, −11.29183760757791, −11.18749066470407, −10.71825654012868, −10.18198637908907, −9.588654922270333, −8.967520212670477, −8.585725738570554, −7.967493042654693, −7.524576374377543, −7.105551971297982, −6.203875311493980, −5.683690944123257, −5.391298018861257, −4.997913501115130, −4.305610170587856, −4.068568625818754, −3.241449458804129, −2.610258966398426, −1.915926746247973, −1.283633260161325, 0, 0,
1.283633260161325, 1.915926746247973, 2.610258966398426, 3.241449458804129, 4.068568625818754, 4.305610170587856, 4.997913501115130, 5.391298018861257, 5.683690944123257, 6.203875311493980, 7.105551971297982, 7.524576374377543, 7.967493042654693, 8.585725738570554, 8.967520212670477, 9.588654922270333, 10.18198637908907, 10.71825654012868, 11.18749066470407, 11.29183760757791, 12.21163684558233, 12.63299895326357, 12.87977629194305, 13.25187152959706, 13.91904082399060
