| L(s) = 1 | − 2-s + 4-s − 8-s − 4·13-s + 16-s − 6·17-s + 2·19-s + 3·23-s − 5·25-s + 4·26-s + 6·29-s + 5·31-s − 32-s + 6·34-s + 8·37-s − 2·38-s − 3·41-s + 2·43-s − 3·46-s + 3·47-s + 5·50-s − 4·52-s − 6·53-s − 6·58-s − 12·59-s + 8·61-s − 5·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.458·19-s + 0.625·23-s − 25-s + 0.784·26-s + 1.11·29-s + 0.898·31-s − 0.176·32-s + 1.02·34-s + 1.31·37-s − 0.324·38-s − 0.468·41-s + 0.304·43-s − 0.442·46-s + 0.437·47-s + 0.707·50-s − 0.554·52-s − 0.824·53-s − 0.787·58-s − 1.56·59-s + 1.02·61-s − 0.635·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.70641597615785949540548779419, −6.68022051339136345214891757881, −6.51780698593554911352687374982, −5.37984989976144824777686999068, −4.72013165165039926172762555214, −3.92935985191376889263274894944, −2.74313502640755318052074716163, −2.31905987356882469481740973903, −1.12322829824585358788849850809, 0,
1.12322829824585358788849850809, 2.31905987356882469481740973903, 2.74313502640755318052074716163, 3.92935985191376889263274894944, 4.72013165165039926172762555214, 5.37984989976144824777686999068, 6.51780698593554911352687374982, 6.68022051339136345214891757881, 7.70641597615785949540548779419
