| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s − 2·9-s − 12-s − 4·13-s + 14-s + 16-s + 2·18-s + 4·19-s + 21-s + 24-s + 4·26-s + 5·27-s − 28-s + 6·29-s − 10·31-s − 32-s − 2·36-s − 8·37-s − 4·38-s + 4·39-s + 3·41-s − 42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.471·18-s + 0.917·19-s + 0.218·21-s + 0.204·24-s + 0.784·26-s + 0.962·27-s − 0.188·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s − 1/3·36-s − 1.31·37-s − 0.648·38-s + 0.640·39-s + 0.468·41-s − 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5180854193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5180854193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−8.160123613927042851473304754479, −7.17503460297761610421202248998, −6.93808090613830957341202849740, −5.89554395447996870335668208962, −5.40629618044085916943448333230, −4.62432042430560900586469318032, −3.38512404447555772422687381402, −2.75703331229368290504728567072, −1.68642953183967073673145648127, −0.42322726168966707272632548378,
0.42322726168966707272632548378, 1.68642953183967073673145648127, 2.75703331229368290504728567072, 3.38512404447555772422687381402, 4.62432042430560900586469318032, 5.40629618044085916943448333230, 5.89554395447996870335668208962, 6.93808090613830957341202849740, 7.17503460297761610421202248998, 8.160123613927042851473304754479
