| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s + 9-s − 3·11-s − 12-s + 2·14-s + 16-s − 18-s + 8·19-s + 2·21-s + 3·22-s − 9·23-s + 24-s − 27-s − 2·28-s + 6·29-s − 4·31-s − 32-s + 3·33-s + 36-s − 11·37-s − 8·38-s − 2·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.534·14-s + 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.436·21-s + 0.639·22-s − 1.87·23-s + 0.204·24-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s − 1.80·37-s − 1.29·38-s − 0.308·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−15.80259823579983, −15.55694405851207, −14.51415974245030, −13.90400634353483, −13.59190197700787, −12.69185339301141, −12.29526741231881, −11.88443546208147, −11.23873095696631, −10.54923842185900, −10.15341457161312, −9.768109049336156, −9.106692585628778, −8.483578180390737, −7.726339694534368, −7.354260970491779, −6.776039821812034, −5.907277324467968, −5.678082263499283, −4.927426649077754, −4.018998630426781, −3.300953152625762, −2.625905540704519, −1.783895868359487, −0.8100856685843732, 0,
0.8100856685843732, 1.783895868359487, 2.625905540704519, 3.300953152625762, 4.018998630426781, 4.927426649077754, 5.678082263499283, 5.907277324467968, 6.776039821812034, 7.354260970491779, 7.726339694534368, 8.483578180390737, 9.106692585628778, 9.768109049336156, 10.15341457161312, 10.54923842185900, 11.23873095696631, 11.88443546208147, 12.29526741231881, 12.69185339301141, 13.59190197700787, 13.90400634353483, 14.51415974245030, 15.55694405851207, 15.80259823579983
