| L(s) = 1 | + 2-s − 3·3-s + 4-s − 4·5-s − 3·6-s − 2·7-s + 8-s + 6·9-s − 4·10-s − 5·11-s − 3·12-s + 13-s − 2·14-s + 12·15-s + 16-s + 2·17-s + 6·18-s − 4·20-s + 6·21-s − 5·22-s + 4·23-s − 3·24-s + 11·25-s + 26-s − 9·27-s − 2·28-s − 8·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s − 1.22·6-s − 0.755·7-s + 0.353·8-s + 2·9-s − 1.26·10-s − 1.50·11-s − 0.866·12-s + 0.277·13-s − 0.534·14-s + 3.09·15-s + 1/4·16-s + 0.485·17-s + 1.41·18-s − 0.894·20-s + 1.30·21-s − 1.06·22-s + 0.834·23-s − 0.612·24-s + 11/5·25-s + 0.196·26-s − 1.73·27-s − 0.377·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 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
−6.96939178969444401091652865832, −6.29366322979871883213658034862, −5.36959841430969789993766396024, −5.13799535567593398514796468067, −4.36136675370327813359263229683, −3.52639876364716764701280346074, −3.08867558938456631625680194517, −1.49693495906042891279298154951, 0, 0,
1.49693495906042891279298154951, 3.08867558938456631625680194517, 3.52639876364716764701280346074, 4.36136675370327813359263229683, 5.13799535567593398514796468067, 5.36959841430969789993766396024, 6.29366322979871883213658034862, 6.96939178969444401091652865832
