| L(s) = 1 | − 3·3-s − 5-s + 6·9-s + 11-s + 6·13-s + 3·15-s − 4·17-s − 6·19-s + 3·23-s − 4·25-s − 9·27-s + 4·29-s − 9·31-s − 3·33-s − 7·37-s − 18·39-s − 2·41-s − 6·43-s − 6·45-s + 12·47-s − 7·49-s + 12·51-s − 2·53-s − 55-s + 18·57-s − 9·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 2·9-s + 0.301·11-s + 1.66·13-s + 0.774·15-s − 0.970·17-s − 1.37·19-s + 0.625·23-s − 4/5·25-s − 1.73·27-s + 0.742·29-s − 1.61·31-s − 0.522·33-s − 1.15·37-s − 2.88·39-s − 0.312·41-s − 0.914·43-s − 0.894·45-s + 1.75·47-s − 49-s + 1.68·51-s − 0.274·53-s − 0.134·55-s + 2.38·57-s − 1.17·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−10.52002101501919326240870090206, −9.160498659687230466098193331552, −8.311467161982721001864529099848, −6.98504509565729693752228882014, −6.40504921502263237245950729880, −5.62850913863761469090185038771, −4.54436529233154883694790452663, −3.74572472636749522129925113169, −1.57722880872445814013578262149, 0,
1.57722880872445814013578262149, 3.74572472636749522129925113169, 4.54436529233154883694790452663, 5.62850913863761469090185038771, 6.40504921502263237245950729880, 6.98504509565729693752228882014, 8.311467161982721001864529099848, 9.160498659687230466098193331552, 10.52002101501919326240870090206
