| L(s) = 1 | + 3-s − 3·5-s + 9-s − 5·11-s + 13-s − 3·15-s − 8·19-s − 2·23-s + 4·25-s + 27-s + 9·29-s − 5·31-s − 5·33-s − 6·37-s + 39-s − 2·41-s + 2·43-s − 3·45-s + 8·47-s − 7·49-s + 53-s + 15·55-s − 8·57-s − 9·59-s − 6·61-s − 3·65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 1.50·11-s + 0.277·13-s − 0.774·15-s − 1.83·19-s − 0.417·23-s + 4/5·25-s + 0.192·27-s + 1.67·29-s − 0.898·31-s − 0.870·33-s − 0.986·37-s + 0.160·39-s − 0.312·41-s + 0.304·43-s − 0.447·45-s + 1.16·47-s − 49-s + 0.137·53-s + 2.02·55-s − 1.05·57-s − 1.17·59-s − 0.768·61-s − 0.372·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.53633705084005, −12.75642508579166, −12.47426748101642, −12.18478960491844, −11.51060827155102, −10.81094491551711, −10.68900253310252, −10.28238811971093, −9.568197644310726, −8.906911853515460, −8.492579166265828, −8.077006879949911, −7.869792445130969, −7.196919625818748, −6.784466226007397, −6.148605866921669, −5.535149242196575, −4.819172177132830, −4.430434201588524, −3.973810896461757, −3.320708293632964, −2.888649500796152, −2.241316065595900, −1.664174289217390, −0.5831776231074953, 0,
0.5831776231074953, 1.664174289217390, 2.241316065595900, 2.888649500796152, 3.320708293632964, 3.973810896461757, 4.430434201588524, 4.819172177132830, 5.535149242196575, 6.148605866921669, 6.784466226007397, 7.196919625818748, 7.869792445130969, 8.077006879949911, 8.492579166265828, 8.906911853515460, 9.568197644310726, 10.28238811971093, 10.68900253310252, 10.81094491551711, 11.51060827155102, 12.18478960491844, 12.47426748101642, 12.75642508579166, 13.53633705084005
