| L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s − 11-s − 13-s + 4·14-s + 16-s − 2·17-s + 4·19-s + 20-s + 22-s + 25-s + 26-s − 4·28-s − 2·29-s + 8·31-s − 32-s + 2·34-s − 4·35-s − 6·37-s − 4·38-s − 40-s + 6·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 0.676·35-s − 0.986·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 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.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−16.46468008703912, −16.09701269564432, −15.64064881444061, −15.05651863549642, −14.29224237971389, −13.59254425200457, −13.18407979861649, −12.62422802452167, −11.92674449633213, −11.46250289221720, −10.52398522625473, −10.09022345089196, −9.743293503640888, −9.037521144720020, −8.628173697155155, −7.688021648909567, −7.120090896740048, −6.554396644956958, −5.968310933677756, −5.327873773771976, −4.389400874809105, −3.393864025818160, −2.861721449420555, −2.120195875081628, −0.9825866260397394, 0,
0.9825866260397394, 2.120195875081628, 2.861721449420555, 3.393864025818160, 4.389400874809105, 5.327873773771976, 5.968310933677756, 6.554396644956958, 7.120090896740048, 7.688021648909567, 8.628173697155155, 9.037521144720020, 9.743293503640888, 10.09022345089196, 10.52398522625473, 11.46250289221720, 11.92674449633213, 12.62422802452167, 13.18407979861649, 13.59254425200457, 14.29224237971389, 15.05651863549642, 15.64064881444061, 16.09701269564432, 16.46468008703912
