| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 4·7-s − 8-s + 9-s − 4·11-s − 12-s + 2·13-s + 4·14-s + 16-s + 2·17-s − 18-s − 19-s + 4·21-s + 4·22-s + 8·23-s + 24-s − 2·26-s − 27-s − 4·28-s + 6·29-s + 4·31-s − 32-s + 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.872·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−8.424191733926246675640608898571, −7.68210243280282842646408090808, −6.76011464101421078127674608335, −6.34412577628387629359364040514, −5.50198314718409406405972989253, −4.58700919259485571170731352212, −3.23152712333822717439104363252, −2.75056461237209541081175930756, −1.13414164856423095138492661577, 0,
1.13414164856423095138492661577, 2.75056461237209541081175930756, 3.23152712333822717439104363252, 4.58700919259485571170731352212, 5.50198314718409406405972989253, 6.34412577628387629359364040514, 6.76011464101421078127674608335, 7.68210243280282842646408090808, 8.424191733926246675640608898571
