| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 2·9-s + 2·12-s + 2·13-s − 4·16-s − 17-s − 4·18-s − 19-s + 23-s + 4·26-s − 5·27-s + 8·31-s − 8·32-s − 2·34-s − 4·36-s + 5·37-s − 2·38-s + 2·39-s − 4·41-s + 10·43-s + 2·46-s − 10·47-s − 4·48-s − 7·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 2/3·9-s + 0.577·12-s + 0.554·13-s − 16-s − 0.242·17-s − 0.942·18-s − 0.229·19-s + 0.208·23-s + 0.784·26-s − 0.962·27-s + 1.43·31-s − 1.41·32-s − 0.342·34-s − 2/3·36-s + 0.821·37-s − 0.324·38-s + 0.320·39-s − 0.624·41-s + 1.52·43-s + 0.294·46-s − 1.45·47-s − 0.577·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−14.73085453634706, −14.07288701946754, −13.76497245258798, −13.39184601135939, −12.80999486426905, −12.39678650413501, −11.68934456253618, −11.37889990906895, −10.89103091450874, −10.13369184566724, −9.494152860666522, −8.944631644528247, −8.458252872628734, −7.957680708718271, −7.243678562082568, −6.513008387010088, −6.095995644881016, −5.672538327868866, −4.807552211760248, −4.542566272821256, −3.702424929815898, −3.323092550511577, −2.639939769305830, −2.215507216587789, −1.157224496147131, 0,
1.157224496147131, 2.215507216587789, 2.639939769305830, 3.323092550511577, 3.702424929815898, 4.542566272821256, 4.807552211760248, 5.672538327868866, 6.095995644881016, 6.513008387010088, 7.243678562082568, 7.957680708718271, 8.458252872628734, 8.944631644528247, 9.494152860666522, 10.13369184566724, 10.89103091450874, 11.37889990906895, 11.68934456253618, 12.39678650413501, 12.80999486426905, 13.39184601135939, 13.76497245258798, 14.07288701946754, 14.73085453634706
