| L(s) = 1 | + 5-s − 11-s + 5·13-s − 5·19-s + 6·23-s + 25-s + 31-s − 7·37-s + 43-s − 12·47-s − 6·53-s − 55-s − 6·59-s − 10·61-s + 5·65-s + 13·67-s − 12·71-s + 5·73-s + 79-s − 5·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·115-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 1.38·13-s − 1.14·19-s + 1.25·23-s + 1/5·25-s + 0.179·31-s − 1.15·37-s + 0.152·43-s − 1.75·47-s − 0.824·53-s − 0.134·55-s − 0.781·59-s − 1.28·61-s + 0.620·65-s + 1.58·67-s − 1.42·71-s + 0.585·73-s + 0.112·79-s − 0.512·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.559·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.92666635969308, −12.29443134940476, −11.72101446847500, −11.25064266849782, −10.79157527526287, −10.56235491669441, −10.09353296369684, −9.360173692628870, −9.119957089336121, −8.615283417040883, −8.168798314937339, −7.815066190463230, −7.002199815491535, −6.690377882098147, −6.222849763566244, −5.827181389177364, −5.202283679808891, −4.751152757336243, −4.298922813687288, −3.515541066833182, −3.244028778134473, −2.628439591226326, −1.843427030929954, −1.541811920436059, −0.7952945146864667, 0,
0.7952945146864667, 1.541811920436059, 1.843427030929954, 2.628439591226326, 3.244028778134473, 3.515541066833182, 4.298922813687288, 4.751152757336243, 5.202283679808891, 5.827181389177364, 6.222849763566244, 6.690377882098147, 7.002199815491535, 7.815066190463230, 8.168798314937339, 8.615283417040883, 9.119957089336121, 9.360173692628870, 10.09353296369684, 10.56235491669441, 10.79157527526287, 11.25064266849782, 11.72101446847500, 12.29443134940476, 12.92666635969308
