| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 7-s − 2·10-s − 5·11-s − 13-s + 2·14-s − 4·16-s + 7·17-s − 6·19-s + 2·20-s + 10·22-s − 3·23-s + 25-s + 2·26-s − 2·28-s − 2·29-s + 2·31-s + 8·32-s − 14·34-s − 35-s + 7·37-s + 12·38-s − 9·41-s − 8·43-s − 10·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 0.632·10-s − 1.50·11-s − 0.277·13-s + 0.534·14-s − 16-s + 1.69·17-s − 1.37·19-s + 0.447·20-s + 2.13·22-s − 0.625·23-s + 1/5·25-s + 0.392·26-s − 0.377·28-s − 0.371·29-s + 0.359·31-s + 1.41·32-s − 2.40·34-s − 0.169·35-s + 1.15·37-s + 1.94·38-s − 1.40·41-s − 1.21·43-s − 1.50·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−10.03210824623968341800708629459, −9.664619125780354113128944873936, −8.333107995317450905784282789603, −7.997850874518894594264893149223, −6.93279974611194334514178729187, −5.86662536743360060967332495720, −4.75380031772719504345197730313, −3.02709857296590648239002053477, −1.78483850539275904424541360817, 0,
1.78483850539275904424541360817, 3.02709857296590648239002053477, 4.75380031772719504345197730313, 5.86662536743360060967332495720, 6.93279974611194334514178729187, 7.997850874518894594264893149223, 8.333107995317450905784282789603, 9.664619125780354113128944873936, 10.03210824623968341800708629459
