| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 12-s + 13-s + 14-s + 15-s + 16-s − 5·17-s + 18-s − 19-s + 20-s + 21-s + 3·23-s + 24-s − 4·25-s + 26-s + 27-s + 28-s − 5·29-s + 30-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 + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.625·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.928·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.968609843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.968609843\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.96537654440623, −13.83185347667958, −13.30168245060559, −12.84908090538941, −12.33447092775809, −11.72238619264625, −11.19116398849408, −10.70213436451882, −10.31551285159786, −9.480522498246562, −9.103448871466908, −8.629237490695661, −7.937096152040470, −7.436819295631093, −6.909983364426229, −6.265886879500726, −5.803334200210926, −5.183064607995325, −4.480762698125622, −4.119673461766579, −3.435936981382881, −2.711651817372737, −2.137388249646997, −1.670881378901017, −0.6763302861485115,
0.6763302861485115, 1.670881378901017, 2.137388249646997, 2.711651817372737, 3.435936981382881, 4.119673461766579, 4.480762698125622, 5.183064607995325, 5.803334200210926, 6.265886879500726, 6.909983364426229, 7.436819295631093, 7.937096152040470, 8.629237490695661, 9.103448871466908, 9.480522498246562, 10.31551285159786, 10.70213436451882, 11.19116398849408, 11.72238619264625, 12.33447092775809, 12.84908090538941, 13.30168245060559, 13.83185347667958, 13.96537654440623
