| L(s) = 1 | + 2-s + 4-s + 3·5-s + 7-s + 8-s − 3·9-s + 3·10-s − 2·13-s + 14-s + 16-s − 2·17-s − 3·18-s + 2·19-s + 3·20-s + 2·23-s + 4·25-s − 2·26-s + 28-s − 4·29-s + 31-s + 32-s − 2·34-s + 3·35-s − 3·36-s − 3·37-s + 2·38-s + 3·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s − 9-s + 0.948·10-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.458·19-s + 0.670·20-s + 0.417·23-s + 4/5·25-s − 0.392·26-s + 0.188·28-s − 0.742·29-s + 0.179·31-s + 0.176·32-s − 0.342·34-s + 0.507·35-s − 1/2·36-s − 0.493·37-s + 0.324·38-s + 0.474·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.290696931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.290696931\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 167 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−11.55949815919557693107925416985, −10.81089896417309785444612051773, −9.735295958456623579159922515476, −8.894431184430811545982274317965, −7.62770548087456982906508608972, −6.39929741517579351282216438404, −5.60760027996819067450043991244, −4.77103237937770595531354209157, −3.06399258007203755791960793021, −1.93996763984281366150661874327,
1.93996763984281366150661874327, 3.06399258007203755791960793021, 4.77103237937770595531354209157, 5.60760027996819067450043991244, 6.39929741517579351282216438404, 7.62770548087456982906508608972, 8.894431184430811545982274317965, 9.735295958456623579159922515476, 10.81089896417309785444612051773, 11.55949815919557693107925416985
