| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s + 11-s − 12-s + 3·14-s + 16-s + 5·17-s + 18-s − 19-s − 3·21-s + 22-s − 3·23-s − 24-s − 27-s + 3·28-s + 2·29-s − 6·31-s + 32-s − 33-s + 5·34-s + 36-s + 3·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.801·14-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 0.229·19-s − 0.654·21-s + 0.213·22-s − 0.625·23-s − 0.204·24-s − 0.192·27-s + 0.566·28-s + 0.371·29-s − 1.07·31-s + 0.176·32-s − 0.174·33-s + 0.857·34-s + 1/6·36-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.679083260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.679083260\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−9.489789657857333122762707892352, −8.383155186282888579791302033314, −7.66737317715158516555540856881, −6.90539921428856432935154258796, −5.84511815958380437455003692156, −5.34821434610320217102293580896, −4.43471661439712863635057431375, −3.66663123638046994304458528693, −2.27390251266660564599788993596, −1.16167432076173546905736506109,
1.16167432076173546905736506109, 2.27390251266660564599788993596, 3.66663123638046994304458528693, 4.43471661439712863635057431375, 5.34821434610320217102293580896, 5.84511815958380437455003692156, 6.90539921428856432935154258796, 7.66737317715158516555540856881, 8.383155186282888579791302033314, 9.489789657857333122762707892352
