| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 11-s − 13-s − 4·14-s + 16-s + 7·17-s + 8·19-s + 22-s − 9·23-s − 26-s − 4·28-s − 6·29-s + 32-s + 7·34-s − 11·37-s + 8·38-s − 2·41-s + 2·43-s + 44-s − 9·46-s + 8·47-s + 9·49-s − 52-s − 9·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 0.301·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.69·17-s + 1.83·19-s + 0.213·22-s − 1.87·23-s − 0.196·26-s − 0.755·28-s − 1.11·29-s + 0.176·32-s + 1.20·34-s − 1.80·37-s + 1.29·38-s − 0.312·41-s + 0.304·43-s + 0.150·44-s − 1.32·46-s + 1.16·47-s + 9/7·49-s − 0.138·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 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.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 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
−14.28511167253470, −13.94876127241979, −13.66125810620399, −12.85162948875204, −12.46683288514088, −12.13816268198900, −11.66622668858881, −11.10309799824051, −10.16141291480058, −9.896447269084975, −9.715388113521891, −8.930983258275800, −8.208695931960156, −7.487010722903522, −7.257879725842096, −6.580925726026295, −5.979142752927031, −5.442142644862246, −5.251805948842613, −4.057218549247820, −3.652478069352071, −3.302794089501224, −2.631382835933571, −1.801079311650731, −0.9771227883570064, 0,
0.9771227883570064, 1.801079311650731, 2.631382835933571, 3.302794089501224, 3.652478069352071, 4.057218549247820, 5.251805948842613, 5.442142644862246, 5.979142752927031, 6.580925726026295, 7.257879725842096, 7.487010722903522, 8.208695931960156, 8.930983258275800, 9.715388113521891, 9.896447269084975, 10.16141291480058, 11.10309799824051, 11.66622668858881, 12.13816268198900, 12.46683288514088, 12.85162948875204, 13.66125810620399, 13.94876127241979, 14.28511167253470
