| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 11-s + 13-s + 16-s + 17-s + 3·18-s − 5·19-s − 22-s + 2·23-s − 5·25-s − 26-s − 5·29-s + 8·31-s − 32-s − 34-s − 3·36-s − 8·37-s + 5·38-s − 6·43-s + 44-s − 2·46-s − 11·47-s + 5·50-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 0.301·11-s + 0.277·13-s + 1/4·16-s + 0.242·17-s + 0.707·18-s − 1.14·19-s − 0.213·22-s + 0.417·23-s − 25-s − 0.196·26-s − 0.928·29-s + 1.43·31-s − 0.176·32-s − 0.171·34-s − 1/2·36-s − 1.31·37-s + 0.811·38-s − 0.914·43-s + 0.150·44-s − 0.294·46-s − 1.60·47-s + 0.707·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−9.148455860909682838797173901070, −8.497896543609873926125954110851, −7.892623211107852747063289943971, −6.77787793409627877002812525492, −6.11487072062742008375174810943, −5.17911252335806779633942989720, −3.88526933294650325440678641270, −2.84543042016155242145018415800, −1.67048910195037592964188302297, 0,
1.67048910195037592964188302297, 2.84543042016155242145018415800, 3.88526933294650325440678641270, 5.17911252335806779633942989720, 6.11487072062742008375174810943, 6.77787793409627877002812525492, 7.892623211107852747063289943971, 8.497896543609873926125954110851, 9.148455860909682838797173901070
