| L(s) = 1 | − 2·5-s − 4·7-s − 6·11-s + 2·13-s − 6·19-s + 4·23-s − 25-s − 29-s + 6·31-s + 8·35-s − 2·37-s + 2·41-s + 10·43-s + 2·47-s + 9·49-s − 10·53-s + 12·55-s − 10·61-s − 4·65-s − 12·67-s + 8·71-s − 10·73-s + 24·77-s + 6·79-s − 16·83-s − 2·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 1.80·11-s + 0.554·13-s − 1.37·19-s + 0.834·23-s − 1/5·25-s − 0.185·29-s + 1.07·31-s + 1.35·35-s − 0.328·37-s + 0.312·41-s + 1.52·43-s + 0.291·47-s + 9/7·49-s − 1.37·53-s + 1.61·55-s − 1.28·61-s − 0.496·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 2.73·77-s + 0.675·79-s − 1.75·83-s − 0.211·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−12.89118537323507, −12.48139332034799, −12.22607845319818, −11.48380025641190, −10.89015326947889, −10.74132497316272, −10.23868968323267, −9.754415025204146, −9.216196332876799, −8.717632358503122, −8.280903563541261, −7.773547640857124, −7.382357978605093, −6.902590100980185, −6.197622376225762, −6.029789212516265, −5.412198942488854, −4.637017094076386, −4.355844534707674, −3.702819334680810, −3.150094122431560, −2.768789931857936, −2.312381346635989, −1.341839392091711, −0.4421936010160557, 0,
0.4421936010160557, 1.341839392091711, 2.312381346635989, 2.768789931857936, 3.150094122431560, 3.702819334680810, 4.355844534707674, 4.637017094076386, 5.412198942488854, 6.029789212516265, 6.197622376225762, 6.902590100980185, 7.382357978605093, 7.773547640857124, 8.280903563541261, 8.717632358503122, 9.216196332876799, 9.754415025204146, 10.23868968323267, 10.74132497316272, 10.89015326947889, 11.48380025641190, 12.22607845319818, 12.48139332034799, 12.89118537323507
