| L(s) = 1 | + 3-s − 7-s + 9-s − 2·11-s + 13-s − 2·17-s + 4·19-s − 21-s + 6·23-s + 27-s + 6·29-s − 4·31-s − 2·33-s − 10·37-s + 39-s + 2·41-s − 10·43-s − 6·47-s + 49-s − 2·51-s + 12·53-s + 4·57-s − 4·61-s − 63-s + 6·69-s − 4·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s − 1.64·37-s + 0.160·39-s + 0.312·41-s − 1.52·43-s − 0.875·47-s + 1/7·49-s − 0.280·51-s + 1.64·53-s + 0.529·57-s − 0.512·61-s − 0.125·63-s + 0.722·69-s − 0.474·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 218400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.25645106343165, −12.95772693666040, −12.27435248366627, −11.86047989318697, −11.39073548509605, −10.78478452277452, −10.28285384777516, −10.07965892100053, −9.362544646952596, −8.866926089182834, −8.675215993042355, −8.027094310032287, −7.503722774988037, −7.008418888637605, −6.679994955806786, −6.044211595209849, −5.318674654424896, −5.020193512045006, −4.466272777046997, −3.647063622870384, −3.301598498435585, −2.839813290835643, −2.185339140180300, −1.534147104204752, −0.8507508079699349, 0,
0.8507508079699349, 1.534147104204752, 2.185339140180300, 2.839813290835643, 3.301598498435585, 3.647063622870384, 4.466272777046997, 5.020193512045006, 5.318674654424896, 6.044211595209849, 6.679994955806786, 7.008418888637605, 7.503722774988037, 8.027094310032287, 8.675215993042355, 8.866926089182834, 9.362544646952596, 10.07965892100053, 10.28285384777516, 10.78478452277452, 11.39073548509605, 11.86047989318697, 12.27435248366627, 12.95772693666040, 13.25645106343165
