| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 4·17-s + 18-s − 19-s + 20-s + 21-s − 2·22-s − 4·23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.872948114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.872948114\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 83 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.78117602516446, −14.39111110999202, −14.18309622068262, −13.52346327004297, −12.93066979261551, −12.54116982772423, −12.09280956111144, −11.30636046577544, −10.80945009465002, −10.26399855006869, −9.674522095236526, −9.248419393941601, −8.340312292503092, −7.919031727467607, −7.500532564788119, −6.707000682875084, −6.138515123920250, −5.436953702427134, −5.051328940698378, −4.229086650295035, −3.750622283144550, −2.843233409158437, −2.438879418962584, −1.701686879208441, −0.7911622996120079,
0.7911622996120079, 1.701686879208441, 2.438879418962584, 2.843233409158437, 3.750622283144550, 4.229086650295035, 5.051328940698378, 5.436953702427134, 6.138515123920250, 6.707000682875084, 7.500532564788119, 7.919031727467607, 8.340312292503092, 9.248419393941601, 9.674522095236526, 10.26399855006869, 10.80945009465002, 11.30636046577544, 12.09280956111144, 12.54116982772423, 12.93066979261551, 13.52346327004297, 14.18309622068262, 14.39111110999202, 14.78117602516446
