| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 13-s − 4·14-s + 16-s + 4·19-s − 20-s + 8·23-s + 25-s − 26-s − 4·28-s + 2·29-s + 8·31-s + 32-s + 4·35-s − 2·37-s + 4·38-s − 40-s − 6·41-s + 12·43-s + 8·46-s + 9·49-s + 50-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 − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.917·19-s − 0.223·20-s + 1.66·23-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 + 0.676·35-s − 0.328·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s + 1.82·43-s + 1.17·46-s + 9/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.83040729314453, −12.35498981533610, −11.96891038146916, −11.69510650501771, −10.96146090444026, −10.61721279196932, −10.14804503414055, −9.606365709014304, −9.197058994671664, −8.851877579536835, −7.966936545262967, −7.769980462239521, −7.012957247692996, −6.712197547130596, −6.453616390463429, −5.680881934228606, −5.344974108238625, −4.686551837525394, −4.332728410006949, −3.539465472368203, −3.287117431951712, −2.773509016339646, −2.396330167084368, −1.292152375094565, −0.8280763793443204, 0,
0.8280763793443204, 1.292152375094565, 2.396330167084368, 2.773509016339646, 3.287117431951712, 3.539465472368203, 4.332728410006949, 4.686551837525394, 5.344974108238625, 5.680881934228606, 6.453616390463429, 6.712197547130596, 7.012957247692996, 7.769980462239521, 7.966936545262967, 8.851877579536835, 9.197058994671664, 9.606365709014304, 10.14804503414055, 10.61721279196932, 10.96146090444026, 11.69510650501771, 11.96891038146916, 12.35498981533610, 12.83040729314453
