| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 4·13-s + 14-s + 16-s + 17-s + 2·19-s − 20-s − 6·23-s + 25-s − 4·26-s + 28-s − 4·31-s + 32-s + 34-s − 35-s + 8·37-s + 2·38-s − 40-s + 6·41-s − 4·43-s − 6·46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.458·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.169·35-s + 1.31·37-s + 0.324·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s − 0.884·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−16.77296535614915, −16.07644854433459, −15.72530732760557, −14.94898336065491, −14.43580107967117, −14.24045364082650, −13.32203107796731, −12.81279029388290, −12.15253138535625, −11.79999036269501, −11.15183740622051, −10.59688108254004, −9.698166490480658, −9.440264325346982, −8.196229966580982, −7.923859684353655, −7.243811138260953, −6.587378655034219, −5.757570991661869, −5.190551810947753, −4.458433188765135, −3.923851598402617, −3.022954255948121, −2.319223871217576, −1.365851168858128, 0,
1.365851168858128, 2.319223871217576, 3.022954255948121, 3.923851598402617, 4.458433188765135, 5.190551810947753, 5.757570991661869, 6.587378655034219, 7.243811138260953, 7.923859684353655, 8.196229966580982, 9.440264325346982, 9.698166490480658, 10.59688108254004, 11.15183740622051, 11.79999036269501, 12.15253138535625, 12.81279029388290, 13.32203107796731, 14.24045364082650, 14.43580107967117, 14.94898336065491, 15.72530732760557, 16.07644854433459, 16.77296535614915
