| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s + 12-s + 4·14-s + 16-s − 18-s − 4·19-s − 4·21-s − 24-s − 5·25-s + 27-s − 4·28-s + 10·31-s − 32-s + 36-s − 2·37-s + 4·38-s − 6·41-s + 4·42-s + 10·43-s + 48-s + 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.872·21-s − 0.204·24-s − 25-s + 0.192·27-s − 0.755·28-s + 1.79·31-s − 0.176·32-s + 1/6·36-s − 0.328·37-s + 0.648·38-s − 0.937·41-s + 0.617·42-s + 1.52·43-s + 0.144·48-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.71979451401888, −13.25377389913865, −12.92015579259542, −12.23667403815725, −11.98694757692141, −11.34445553218618, −10.62975017477701, −10.15208163324259, −9.977293056755207, −9.393325305425493, −8.821453819625705, −8.617510546893201, −7.844918005567269, −7.503574393100444, −6.722285076212764, −6.547988089316380, −5.923221577429817, −5.421327043719245, −4.273355683350742, −4.166877137398377, −3.216850944770705, −2.905955120805301, −2.295777708336803, −1.608279446353821, −0.7163829214583452, 0,
0.7163829214583452, 1.608279446353821, 2.295777708336803, 2.905955120805301, 3.216850944770705, 4.166877137398377, 4.273355683350742, 5.421327043719245, 5.923221577429817, 6.547988089316380, 6.722285076212764, 7.503574393100444, 7.844918005567269, 8.617510546893201, 8.821453819625705, 9.393325305425493, 9.977293056755207, 10.15208163324259, 10.62975017477701, 11.34445553218618, 11.98694757692141, 12.23667403815725, 12.92015579259542, 13.25377389913865, 13.71979451401888
