| 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.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 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.ac |
| 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.g |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 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.m |
| 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.ag |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−16.23382584343725, −15.79237468490892, −14.98938492702851, −14.56112754623434, −13.78866987171594, −13.48643581653699, −12.89385713189876, −12.33270178554292, −11.96133405874991, −11.03876799549318, −10.31804180366246, −9.992945792928853, −9.400840343167048, −9.104167574820993, −8.174766759275462, −7.437267664932844, −6.877843863884080, −6.223618040690054, −5.701028209707070, −5.250012727220676, −4.051881733639938, −3.672383910389034, −2.594237044857750, −2.297437106945966, −1.067927774656702, 0,
1.067927774656702, 2.297437106945966, 2.594237044857750, 3.672383910389034, 4.051881733639938, 5.250012727220676, 5.701028209707070, 6.223618040690054, 6.877843863884080, 7.437267664932844, 8.174766759275462, 9.104167574820993, 9.400840343167048, 9.992945792928853, 10.31804180366246, 11.03876799549318, 11.96133405874991, 12.33270178554292, 12.89385713189876, 13.48643581653699, 13.78866987171594, 14.56112754623434, 14.98938492702851, 15.79237468490892, 16.23382584343725
