| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s − 5·13-s + 16-s − 18-s − 7·19-s − 3·23-s + 2·24-s + 5·26-s + 4·27-s − 3·29-s − 5·31-s − 32-s + 36-s + 4·37-s + 7·38-s + 10·39-s + 12·41-s + 5·43-s + 3·46-s − 2·48-s − 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.38·13-s + 1/4·16-s − 0.235·18-s − 1.60·19-s − 0.625·23-s + 0.408·24-s + 0.980·26-s + 0.769·27-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 1/6·36-s + 0.657·37-s + 1.13·38-s + 1.60·39-s + 1.87·41-s + 0.762·43-s + 0.442·46-s − 0.288·48-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−12.54863669191939, −12.41872392867409, −12.07603422440319, −11.39206582801088, −11.03242910738315, −10.62474067076421, −10.42039931767810, −9.654153100138103, −9.279315058235375, −8.988455876352461, −8.189753188199479, −7.763483426699550, −7.389935559832285, −6.861009187520006, −6.228382478337322, −5.972057325376640, −5.574273838311195, −4.745762597093312, −4.516964252734163, −3.924569491218526, −3.012752290545828, −2.526024741906553, −1.967306678121948, −1.332781958123584, −0.4290950027997603, 0,
0.4290950027997603, 1.332781958123584, 1.967306678121948, 2.526024741906553, 3.012752290545828, 3.924569491218526, 4.516964252734163, 4.745762597093312, 5.574273838311195, 5.972057325376640, 6.228382478337322, 6.861009187520006, 7.389935559832285, 7.763483426699550, 8.189753188199479, 8.988455876352461, 9.279315058235375, 9.654153100138103, 10.42039931767810, 10.62474067076421, 11.03242910738315, 11.39206582801088, 12.07603422440319, 12.41872392867409, 12.54863669191939
