| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s − 3·11-s + 4·13-s + 15-s − 5·19-s + 3·21-s + 25-s + 27-s + 5·29-s − 4·31-s − 3·33-s + 3·35-s − 9·37-s + 4·39-s − 7·41-s − 8·43-s + 45-s − 3·47-s + 2·49-s − 7·53-s − 3·55-s − 5·57-s + 6·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.258·15-s − 1.14·19-s + 0.654·21-s + 1/5·25-s + 0.192·27-s + 0.928·29-s − 0.718·31-s − 0.522·33-s + 0.507·35-s − 1.47·37-s + 0.640·39-s − 1.09·41-s − 1.21·43-s + 0.149·45-s − 0.437·47-s + 2/7·49-s − 0.961·53-s − 0.404·55-s − 0.662·57-s + 0.781·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.20670034699089, −14.62539626071058, −14.16438747600950, −13.72206336585343, −13.12472007870463, −12.84914832612636, −12.05168506655038, −11.46401648758191, −10.89323120028942, −10.40694595227592, −10.08161355241318, −9.167576781333121, −8.611610936741414, −8.287598669754150, −7.891732507822760, −7.005208258006618, −6.544695556444983, −5.820852716008078, −5.044122671008646, −4.804912600215546, −3.870347300986516, −3.314970686439374, −2.473700232795512, −1.804876944826453, −1.341224404838063, 0,
1.341224404838063, 1.804876944826453, 2.473700232795512, 3.314970686439374, 3.870347300986516, 4.804912600215546, 5.044122671008646, 5.820852716008078, 6.544695556444983, 7.005208258006618, 7.891732507822760, 8.287598669754150, 8.611610936741414, 9.167576781333121, 10.08161355241318, 10.40694595227592, 10.89323120028942, 11.46401648758191, 12.05168506655038, 12.84914832612636, 13.12472007870463, 13.72206336585343, 14.16438747600950, 14.62539626071058, 15.20670034699089
