| L(s) = 1 | − 2·4-s − 7-s − 4·11-s + 4·16-s + 4·17-s + 19-s − 4·23-s + 2·28-s − 4·29-s + 5·31-s + 2·37-s − 4·43-s + 8·44-s − 2·47-s + 49-s + 6·53-s − 12·59-s − 15·61-s − 8·64-s + 4·67-s − 8·68-s − 10·71-s + 3·73-s − 2·76-s + 4·77-s − 11·79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s − 1.20·11-s + 16-s + 0.970·17-s + 0.229·19-s − 0.834·23-s + 0.377·28-s − 0.742·29-s + 0.898·31-s + 0.328·37-s − 0.609·43-s + 1.20·44-s − 0.291·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.92·61-s − 64-s + 0.488·67-s − 0.970·68-s − 1.18·71-s + 0.351·73-s − 0.229·76-s + 0.455·77-s − 1.23·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 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 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.24479985039389, −12.78338452361152, −12.52130126520562, −11.87895052699959, −11.59381527435705, −10.70686275947864, −10.46579433892234, −9.889419885226628, −9.726845613067143, −9.100107568945099, −8.601405909922539, −8.112280745735605, −7.651730616366688, −7.443832733409194, −6.567950150795003, −5.986074564381058, −5.627676966532798, −5.152142425033994, −4.628660003215245, −4.109865652097218, −3.559934573812204, −2.950611118431155, −2.620012067756649, −1.606060400940269, −1.117207046839208, 0, 0,
1.117207046839208, 1.606060400940269, 2.620012067756649, 2.950611118431155, 3.559934573812204, 4.109865652097218, 4.628660003215245, 5.152142425033994, 5.627676966532798, 5.986074564381058, 6.567950150795003, 7.443832733409194, 7.651730616366688, 8.112280745735605, 8.601405909922539, 9.100107568945099, 9.726845613067143, 9.889419885226628, 10.46579433892234, 10.70686275947864, 11.59381527435705, 11.87895052699959, 12.52130126520562, 12.78338452361152, 13.24479985039389
