| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s − 2·7-s + 8-s + 9-s − 2·10-s − 12-s − 2·14-s + 2·15-s + 16-s + 18-s − 6·19-s − 2·20-s + 2·21-s + 4·23-s − 24-s − 25-s − 27-s − 2·28-s + 10·29-s + 2·30-s − 10·31-s + 32-s + 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s − 0.534·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.447·20-s + 0.436·21-s + 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.377·28-s + 1.85·29-s + 0.365·30-s − 1.79·31-s + 0.176·32-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.83746372879062, −12.52276580815316, −12.04919926437697, −11.67050334634394, −11.21116183086067, −10.64627924690505, −10.48180991246625, −9.848138313213198, −9.244745202216564, −8.767780972172581, −8.222507246405092, −7.710402778423336, −7.196068239562184, −6.747459035402137, −6.315835177089756, −5.965313223287185, −5.256293939417764, −4.792308299246188, −4.294179285403598, −3.901160771563795, −3.280304832092438, −2.883590715694078, −2.139788031201585, −1.472277636570888, −0.6199758784651995, 0,
0.6199758784651995, 1.472277636570888, 2.139788031201585, 2.883590715694078, 3.280304832092438, 3.901160771563795, 4.294179285403598, 4.792308299246188, 5.256293939417764, 5.965313223287185, 6.315835177089756, 6.747459035402137, 7.196068239562184, 7.710402778423336, 8.222507246405092, 8.767780972172581, 9.244745202216564, 9.848138313213198, 10.48180991246625, 10.64627924690505, 11.21116183086067, 11.67050334634394, 12.04919926437697, 12.52276580815316, 12.83746372879062
