| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 6·11-s − 7·13-s − 2·15-s + 2·17-s + 21-s − 25-s + 27-s + 10·29-s + 8·31-s − 6·33-s − 2·35-s + 7·37-s − 7·39-s + 9·43-s − 2·45-s − 8·47-s − 6·49-s + 2·51-s + 2·53-s + 12·55-s + 4·59-s + 7·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s − 1.94·13-s − 0.516·15-s + 0.485·17-s + 0.218·21-s − 1/5·25-s + 0.192·27-s + 1.85·29-s + 1.43·31-s − 1.04·33-s − 0.338·35-s + 1.15·37-s − 1.12·39-s + 1.37·43-s − 0.298·45-s − 1.16·47-s − 6/7·49-s + 0.280·51-s + 0.274·53-s + 1.61·55-s + 0.520·59-s + 0.896·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−15.59471792931394, −15.02900559612742, −14.64246647656146, −14.13162292878807, −13.48601629002874, −12.83114853700985, −12.49789853099641, −11.74439030834140, −11.55066658458697, −10.52651884897407, −10.10429657110346, −9.837764224972983, −8.908615026233894, −8.166201121454905, −7.866979800668905, −7.563122217736854, −6.887643539407144, −6.014452522899214, −5.139526773700628, −4.704896292130450, −4.288027038460561, −3.169722459520381, −2.693459640862498, −2.244586791113931, −0.9196795965710561, 0,
0.9196795965710561, 2.244586791113931, 2.693459640862498, 3.169722459520381, 4.288027038460561, 4.704896292130450, 5.139526773700628, 6.014452522899214, 6.887643539407144, 7.563122217736854, 7.866979800668905, 8.166201121454905, 8.908615026233894, 9.837764224972983, 10.10429657110346, 10.52651884897407, 11.55066658458697, 11.74439030834140, 12.49789853099641, 12.83114853700985, 13.48601629002874, 14.13162292878807, 14.64246647656146, 15.02900559612742, 15.59471792931394
