| L(s) = 1 | − 2·7-s − 3·11-s + 2·13-s + 3·17-s + 19-s − 6·23-s − 5·25-s − 6·29-s + 4·31-s − 4·37-s − 9·41-s + 43-s − 6·47-s − 3·49-s − 12·53-s + 3·59-s + 8·61-s − 5·67-s − 12·71-s + 11·73-s + 6·77-s + 4·79-s + 12·83-s − 6·89-s − 4·91-s + 5·97-s − 14·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.904·11-s + 0.554·13-s + 0.727·17-s + 0.229·19-s − 1.25·23-s − 25-s − 1.11·29-s + 0.718·31-s − 0.657·37-s − 1.40·41-s + 0.152·43-s − 0.875·47-s − 3/7·49-s − 1.64·53-s + 0.390·59-s + 1.02·61-s − 0.610·67-s − 1.42·71-s + 1.28·73-s + 0.683·77-s + 0.450·79-s + 1.31·83-s − 0.635·89-s − 0.419·91-s + 0.507·97-s − 1.37·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−9.463629439715469903269613948918, −8.255589340941165804909076040404, −7.79453288119723697641112445352, −6.69953527334383655382904548898, −5.91234365712937538559209143490, −5.13777464868847606985261506723, −3.86098888246252906299834258338, −3.10746305536689930220572132603, −1.80773362483917328069437252787, 0,
1.80773362483917328069437252787, 3.10746305536689930220572132603, 3.86098888246252906299834258338, 5.13777464868847606985261506723, 5.91234365712937538559209143490, 6.69953527334383655382904548898, 7.79453288119723697641112445352, 8.255589340941165804909076040404, 9.463629439715469903269613948918
