| L(s) = 1 | − 3-s − 2·7-s + 9-s + 11-s − 4·13-s + 2·17-s + 8·19-s + 2·21-s − 2·23-s − 5·25-s − 27-s + 2·29-s − 8·31-s − 33-s − 37-s + 4·39-s + 6·41-s − 12·43-s − 10·47-s − 3·49-s − 2·51-s − 12·53-s − 8·57-s + 4·59-s − 2·63-s + 12·67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.485·17-s + 1.83·19-s + 0.436·21-s − 0.417·23-s − 25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s − 0.164·37-s + 0.640·39-s + 0.937·41-s − 1.82·43-s − 1.45·47-s − 3/7·49-s − 0.280·51-s − 1.64·53-s − 1.05·57-s + 0.520·59-s − 0.251·63-s + 1.46·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3904044851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3904044851\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.14122221293996, −13.31331723152127, −13.08949270349049, −12.44429021764922, −11.97504503251907, −11.62098821781055, −11.20402286711472, −10.37830126261294, −9.915804397074035, −9.528846259516835, −9.352899121387134, −8.271151941332707, −7.890263434042354, −7.156721516360649, −6.972031180877822, −6.224825661704048, −5.611273943028382, −5.304973168484833, −4.619325761825281, −3.967544052922335, −3.209465280061951, −2.948752728724556, −1.827945672717355, −1.346250798368414, −0.2126326258001332,
0.2126326258001332, 1.346250798368414, 1.827945672717355, 2.948752728724556, 3.209465280061951, 3.967544052922335, 4.619325761825281, 5.304973168484833, 5.611273943028382, 6.224825661704048, 6.972031180877822, 7.156721516360649, 7.890263434042354, 8.271151941332707, 9.352899121387134, 9.528846259516835, 9.915804397074035, 10.37830126261294, 11.20402286711472, 11.62098821781055, 11.97504503251907, 12.44429021764922, 13.08949270349049, 13.31331723152127, 14.14122221293996
