| L(s) = 1 | + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s + 5·11-s − 3·12-s − 13-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s + 19-s + 20-s − 3·21-s + 5·22-s + 3·23-s − 3·24-s + 25-s − 26-s − 9·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 1.50·11-s − 0.866·12-s − 0.277·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s + 0.229·19-s + 0.223·20-s − 0.654·21-s + 1.06·22-s + 0.625·23-s − 0.612·24-s + 1/5·25-s − 0.196·26-s − 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.48876584302175, −13.72143188492278, −13.35975059713122, −12.72433826863630, −12.25621512939526, −11.84012752782409, −11.46923345837574, −11.08372759274709, −10.45699220925492, −10.11027547512542, −9.361724939700115, −8.941752477290005, −8.132474188469215, −7.314827176802090, −6.844004153253039, −6.411806306614508, −6.145053306001535, −5.310647763189589, −5.045512197399587, −4.452688324006917, −3.989077493232467, −3.197769136312896, −2.271214396765305, −1.458106798839374, −1.096184065052373, 0,
1.096184065052373, 1.458106798839374, 2.271214396765305, 3.197769136312896, 3.989077493232467, 4.452688324006917, 5.045512197399587, 5.310647763189589, 6.145053306001535, 6.411806306614508, 6.844004153253039, 7.314827176802090, 8.132474188469215, 8.941752477290005, 9.361724939700115, 10.11027547512542, 10.45699220925492, 11.08372759274709, 11.46923345837574, 11.84012752782409, 12.25621512939526, 12.72433826863630, 13.35975059713122, 13.72143188492278, 14.48876584302175
