| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 4·11-s − 2·13-s − 15-s + 2·17-s − 4·19-s − 21-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·33-s + 35-s − 2·37-s + 2·39-s + 2·41-s − 4·43-s + 45-s + 49-s − 2·51-s − 10·53-s − 4·55-s + 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 0.539·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−9.009611137904859548991349662245, −8.060561717226913839356761311067, −7.38480877575677725511614353546, −6.49541389816732821186277653939, −5.48998134657804488643602211945, −5.10751363306408757224852565788, −4.01092057725677360635799148336, −2.70612473469008799964811139117, −1.69445881682715379624957094269, 0,
1.69445881682715379624957094269, 2.70612473469008799964811139117, 4.01092057725677360635799148336, 5.10751363306408757224852565788, 5.48998134657804488643602211945, 6.49541389816732821186277653939, 7.38480877575677725511614353546, 8.060561717226913839356761311067, 9.009611137904859548991349662245
