| 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 + 6·29-s + 4·31-s + 32-s + 2·34-s + 35-s − 2·37-s + 4·38-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 + 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 + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.328·37-s + 0.648·38-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
| \(L(1)\) |
\(\approx\) |
\(4.686311651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.686311651\) |
| \(L(\frac{3}{2})\) |
|
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.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.890352008265981388580693455500, −6.74017478362533921219216665299, −6.54175771100620190342781826842, −5.65519556198623421107545078598, −5.05909529110962376968688001383, −4.27701978619790032461550090165, −3.57305333487530746729940557808, −2.80077536046019404608493976803, −1.75650564282882643836951396600, −1.05398318131943270328651841713,
1.05398318131943270328651841713, 1.75650564282882643836951396600, 2.80077536046019404608493976803, 3.57305333487530746729940557808, 4.27701978619790032461550090165, 5.05909529110962376968688001383, 5.65519556198623421107545078598, 6.54175771100620190342781826842, 6.74017478362533921219216665299, 7.890352008265981388580693455500
