| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·5-s − 2·6-s − 2·9-s − 4·10-s + 3·11-s − 2·12-s − 6·13-s + 2·15-s − 4·16-s + 2·17-s − 4·18-s + 6·19-s − 4·20-s + 6·22-s − 4·23-s − 25-s − 12·26-s + 5·27-s − 4·29-s + 4·30-s − 8·32-s − 3·33-s + 4·34-s − 4·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s − 2/3·9-s − 1.26·10-s + 0.904·11-s − 0.577·12-s − 1.66·13-s + 0.516·15-s − 16-s + 0.485·17-s − 0.942·18-s + 1.37·19-s − 0.894·20-s + 1.27·22-s − 0.834·23-s − 1/5·25-s − 2.35·26-s + 0.962·27-s − 0.742·29-s + 0.730·30-s − 1.41·32-s − 0.522·33-s + 0.685·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67081 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.34813631681624, −14.10010101787034, −13.51997531310378, −12.80078257863852, −12.27075109993223, −11.89532001080183, −11.78704613950726, −11.31033681081277, −10.64961065706977, −9.822400405338175, −9.490814053796274, −8.840949245418108, −8.062598315034296, −7.556758924953254, −7.050859617175818, −6.526626795617000, −5.728081191358792, −5.527184719161607, −4.892898782135545, −4.394063119983784, −3.741115912067021, −3.349340684411870, −2.654167730909465, −1.968771792695657, −0.7734911904951141, 0,
0.7734911904951141, 1.968771792695657, 2.654167730909465, 3.349340684411870, 3.741115912067021, 4.394063119983784, 4.892898782135545, 5.527184719161607, 5.728081191358792, 6.526626795617000, 7.050859617175818, 7.556758924953254, 8.062598315034296, 8.840949245418108, 9.490814053796274, 9.822400405338175, 10.64961065706977, 11.31033681081277, 11.78704613950726, 11.89532001080183, 12.27075109993223, 12.80078257863852, 13.51997531310378, 14.10010101787034, 14.34813631681624
