| L(s) = 1 | + 2-s − 2·3-s + 4-s + 5-s − 2·6-s − 7-s + 8-s + 9-s + 10-s + 11-s − 2·12-s − 14-s − 2·15-s + 16-s − 6·17-s + 18-s − 2·19-s + 20-s + 2·21-s + 22-s − 6·23-s − 2·24-s + 25-s + 4·27-s − 28-s − 2·30-s − 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.577·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s + 0.213·22-s − 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.769·27-s − 0.188·28-s − 0.365·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.78859020007677, −13.39176282637189, −13.02966684594928, −12.51841903953302, −12.03127346537377, −11.56951190171090, −11.21722331394963, −10.73043679240218, −10.24352200223557, −9.779861653174034, −9.191210962869731, −8.526273032013279, −8.169765421575093, −7.173535369806269, −6.865532419689976, −6.356102427106565, −5.993698032925355, −5.559772393845641, −4.867413576814217, −4.560204457911096, −3.868243458201565, −3.302495440747127, −2.541619397543033, −1.886946507140484, −1.382233245886780, 0, 0,
1.382233245886780, 1.886946507140484, 2.541619397543033, 3.302495440747127, 3.868243458201565, 4.560204457911096, 4.867413576814217, 5.559772393845641, 5.993698032925355, 6.356102427106565, 6.865532419689976, 7.173535369806269, 8.169765421575093, 8.526273032013279, 9.191210962869731, 9.779861653174034, 10.24352200223557, 10.73043679240218, 11.21722331394963, 11.56951190171090, 12.03127346537377, 12.51841903953302, 13.02966684594928, 13.39176282637189, 13.78859020007677
