| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 4·11-s + 13-s − 14-s + 16-s + 6·17-s − 8·19-s + 20-s − 4·22-s − 8·23-s + 25-s + 26-s − 28-s − 2·29-s + 8·31-s + 32-s + 6·34-s − 35-s + 2·37-s − 8·38-s + 40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.83·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s + 0.328·37-s − 1.29·38-s + 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.48216639426499517189374012546, −6.52356162408429174996205154834, −5.96326244067141954922067898963, −5.55353215764026904298583923917, −4.58901717840721965550264142523, −3.99944499214615554905986601028, −3.02076803872126801409506650810, −2.43740410427934545923545943054, −1.49557201818744338694494800248, 0,
1.49557201818744338694494800248, 2.43740410427934545923545943054, 3.02076803872126801409506650810, 3.99944499214615554905986601028, 4.58901717840721965550264142523, 5.55353215764026904298583923917, 5.96326244067141954922067898963, 6.52356162408429174996205154834, 7.48216639426499517189374012546
