| L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s − 7-s − 3·8-s − 2·9-s − 2·10-s − 4·11-s − 12-s + 6·13-s − 14-s − 2·15-s − 16-s + 17-s − 2·18-s − 8·19-s + 2·20-s − 21-s − 4·22-s − 6·23-s − 3·24-s − 25-s + 6·26-s − 5·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 1.06·8-s − 2/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s + 0.242·17-s − 0.471·18-s − 1.83·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s − 1.25·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s − 0.962·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 593 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 593 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 593 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 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 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−10.39528265659794541046283588472, −9.101205623812072394286895499734, −8.363790849736602947801395155266, −7.957216046222589425071841416019, −6.33555495514114102983280626908, −5.63266990976124679184471190359, −4.26904521293272908448861856309, −3.65594645642779513781839626146, −2.61990803623846597107296677552, 0,
2.61990803623846597107296677552, 3.65594645642779513781839626146, 4.26904521293272908448861856309, 5.63266990976124679184471190359, 6.33555495514114102983280626908, 7.957216046222589425071841416019, 8.363790849736602947801395155266, 9.101205623812072394286895499734, 10.39528265659794541046283588472
