| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 6·11-s + 2·12-s + 4·13-s + 14-s + 16-s + 6·17-s + 18-s + 2·19-s + 2·21-s + 6·22-s − 3·23-s + 2·24-s + 4·26-s − 4·27-s + 28-s − 6·29-s + 32-s + 12·33-s + 6·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.436·21-s + 1.27·22-s − 0.625·23-s + 0.408·24-s + 0.784·26-s − 0.769·27-s + 0.188·28-s − 1.11·29-s + 0.176·32-s + 2.08·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.363304591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.363304591\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.45017912807507, −14.20095436209942, −13.68696182808462, −13.27271262189259, −12.49797045077009, −12.07871149457061, −11.56998272033586, −11.09652456114817, −10.51565278760037, −9.645759794086097, −9.271504306008165, −8.916047712461557, −8.109913860711972, −7.736774677665411, −7.277285891458007, −6.390507734290612, −5.924300920277639, −5.536023825990969, −4.439962001214608, −4.061824938856793, −3.426117722938913, −3.172339654839893, −2.179678975076519, −1.531925044522006, −0.9868325066964774,
0.9868325066964774, 1.531925044522006, 2.179678975076519, 3.172339654839893, 3.426117722938913, 4.061824938856793, 4.439962001214608, 5.536023825990969, 5.924300920277639, 6.390507734290612, 7.277285891458007, 7.736774677665411, 8.109913860711972, 8.916047712461557, 9.271504306008165, 9.645759794086097, 10.51565278760037, 11.09652456114817, 11.56998272033586, 12.07871149457061, 12.49797045077009, 13.27271262189259, 13.68696182808462, 14.20095436209942, 14.45017912807507
