| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 4·11-s − 13-s + 14-s + 16-s + 2·17-s − 4·19-s + 20-s − 4·22-s + 4·23-s + 25-s + 26-s − 28-s + 2·29-s − 8·31-s − 32-s − 2·34-s − 35-s − 10·37-s + 4·38-s − 40-s − 6·41-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 + 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 0.834·23-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 − 0.342·34-s − 0.169·35-s − 1.64·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-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.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−7.32173470457478790496818503546, −6.88650619781395565543791595111, −6.26537903298809040970582852825, −5.55156247363261922316694733429, −4.69870468661199304233825873226, −3.71018361737966048609477445498, −3.05425976181833450906296398990, −1.97259854268338008528174805520, −1.30540079953304387560472022924, 0,
1.30540079953304387560472022924, 1.97259854268338008528174805520, 3.05425976181833450906296398990, 3.71018361737966048609477445498, 4.69870468661199304233825873226, 5.55156247363261922316694733429, 6.26537903298809040970582852825, 6.88650619781395565543791595111, 7.32173470457478790496818503546
