| L(s) = 1 | − 2-s + 4-s − 2·5-s − 4·7-s − 8-s + 2·10-s + 2·11-s + 4·14-s + 16-s + 6·17-s − 3·19-s − 2·20-s − 2·22-s − 4·23-s − 25-s − 4·28-s − 4·29-s + 5·31-s − 32-s − 6·34-s + 8·35-s + 3·37-s + 3·38-s + 2·40-s − 6·41-s + 7·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 0.353·8-s + 0.632·10-s + 0.603·11-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.688·19-s − 0.447·20-s − 0.426·22-s − 0.834·23-s − 1/5·25-s − 0.755·28-s − 0.742·29-s + 0.898·31-s − 0.176·32-s − 1.02·34-s + 1.35·35-s + 0.493·37-s + 0.486·38-s + 0.316·40-s − 0.937·41-s + 1.06·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.39817629290458357844831457915, −6.85425688679637173788038592217, −6.13256967104391770121389314835, −5.62915593411303146463645006445, −4.33607216669422279608954374660, −3.66350345752162845202271348890, −3.17434183398426959328104402213, −2.15566173481552240708340121786, −0.919633151002002238213231666565, 0,
0.919633151002002238213231666565, 2.15566173481552240708340121786, 3.17434183398426959328104402213, 3.66350345752162845202271348890, 4.33607216669422279608954374660, 5.62915593411303146463645006445, 6.13256967104391770121389314835, 6.85425688679637173788038592217, 7.39817629290458357844831457915
