| L(s) = 1 | − 2·5-s + 4·7-s − 2·13-s − 2·17-s − 4·23-s − 25-s − 29-s + 6·31-s − 8·35-s + 4·37-s − 2·41-s − 4·43-s + 8·47-s + 9·49-s − 14·53-s + 6·59-s + 8·61-s + 4·65-s + 12·67-s + 16·71-s − 2·73-s − 6·79-s − 2·83-s + 4·85-s − 14·89-s − 8·91-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.554·13-s − 0.485·17-s − 0.834·23-s − 1/5·25-s − 0.185·29-s + 1.07·31-s − 1.35·35-s + 0.657·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 1.92·53-s + 0.781·59-s + 1.02·61-s + 0.496·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s − 0.675·79-s − 0.219·83-s + 0.433·85-s − 1.48·89-s − 0.838·91-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.94496915609257, −15.64090248411543, −15.11699465342758, −14.49028765595426, −14.14971004806163, −13.52401922151572, −12.74790282783279, −12.13060169147974, −11.68264532252429, −11.23854079137267, −10.78818509289418, −9.945018858529290, −9.462037486831537, −8.441892951785678, −8.190854524118834, −7.750001706442361, −7.045255636692068, −6.387828580537805, −5.436491433514295, −4.942646988671114, −4.224058431733923, −3.857858648992019, −2.702256292283991, −2.037341423717585, −1.123451994386529, 0,
1.123451994386529, 2.037341423717585, 2.702256292283991, 3.857858648992019, 4.224058431733923, 4.942646988671114, 5.436491433514295, 6.387828580537805, 7.045255636692068, 7.750001706442361, 8.190854524118834, 8.441892951785678, 9.462037486831537, 9.945018858529290, 10.78818509289418, 11.23854079137267, 11.68264532252429, 12.13060169147974, 12.74790282783279, 13.52401922151572, 14.14971004806163, 14.49028765595426, 15.11699465342758, 15.64090248411543, 15.94496915609257
