| L(s) = 1 | + 2-s − 3·3-s + 4-s − 5-s − 3·6-s + 7-s + 8-s + 6·9-s − 10-s + 11-s − 3·12-s + 13-s + 14-s + 3·15-s + 16-s + 4·17-s + 6·18-s + 6·19-s − 20-s − 3·21-s + 22-s − 4·23-s − 3·24-s − 4·25-s + 26-s − 9·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.316·10-s + 0.301·11-s − 0.866·12-s + 0.277·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s + 1.37·19-s − 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s − 0.612·24-s − 4/5·25-s + 0.196·26-s − 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 239 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−14.28483210540453, −13.89555353498605, −13.43481819053742, −12.64062194277883, −12.25024085406248, −11.90027525375710, −11.49910431890645, −11.24607192396697, −10.46069474401701, −10.02771362592177, −9.739835522178438, −8.697889831827480, −8.010174344856196, −7.620321549773256, −6.871478492074432, −6.591525614349257, −5.904526104817460, −5.352847003838308, −5.143103398355161, −4.464639531707257, −3.770634776384822, −3.437404987072640, −2.366144450781090, −1.406347451858428, −1.001489292903611, 0,
1.001489292903611, 1.406347451858428, 2.366144450781090, 3.437404987072640, 3.770634776384822, 4.464639531707257, 5.143103398355161, 5.352847003838308, 5.904526104817460, 6.591525614349257, 6.871478492074432, 7.620321549773256, 8.010174344856196, 8.697889831827480, 9.739835522178438, 10.02771362592177, 10.46069474401701, 11.24607192396697, 11.49910431890645, 11.90027525375710, 12.25024085406248, 12.64062194277883, 13.43481819053742, 13.89555353498605, 14.28483210540453
