| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 3·7-s − 8-s + 9-s + 2·10-s + 11-s − 12-s + 13-s + 3·14-s + 2·15-s + 16-s − 17-s − 18-s − 4·19-s − 2·20-s + 3·21-s − 22-s + 2·23-s + 24-s − 25-s − 26-s − 27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.654·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 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
−16.42208429895078, −16.04033730610440, −15.36831494384759, −15.00044712132825, −14.29729528951993, −13.32782731982856, −12.79570662632189, −12.51867127899571, −11.64302541989488, −11.27172927084859, −10.79667130359423, −10.07483235350304, −9.513269328576703, −8.959577529768322, −8.354804769025143, −7.531902334308715, −7.146284672273492, −6.412272741783395, −5.989658139632815, −5.170476766807593, −4.033573765314735, −3.826625515914170, −2.835945618855465, −1.929527759069653, −0.7605273698087398, 0,
0.7605273698087398, 1.929527759069653, 2.835945618855465, 3.826625515914170, 4.033573765314735, 5.170476766807593, 5.989658139632815, 6.412272741783395, 7.146284672273492, 7.531902334308715, 8.354804769025143, 8.959577529768322, 9.513269328576703, 10.07483235350304, 10.79667130359423, 11.27172927084859, 11.64302541989488, 12.51867127899571, 12.79570662632189, 13.32782731982856, 14.29729528951993, 15.00044712132825, 15.36831494384759, 16.04033730610440, 16.42208429895078
