| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s − 2·9-s + 2·10-s − 5·11-s − 12-s + 13-s − 2·15-s + 16-s + 8·17-s − 2·18-s + 8·19-s + 2·20-s − 5·22-s + 7·23-s − 24-s − 25-s + 26-s + 5·27-s + 6·29-s − 2·30-s − 5·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 1.50·11-s − 0.288·12-s + 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.94·17-s − 0.471·18-s + 1.83·19-s + 0.447·20-s − 1.06·22-s + 1.45·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.962·27-s + 1.11·29-s − 0.365·30-s − 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.421247775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.421247775\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−10.05673566918767625250193400504, −8.903429236009401349480128932011, −7.83453299422049901459894856291, −7.12954892125448907737267826189, −5.92467186361142389098445547120, −5.39886040244092591091207033663, −5.08193250239214255940378810642, −3.31827057369299548889601930734, −2.71892018094222102094293573657, −1.13853543508304244978172677070,
1.13853543508304244978172677070, 2.71892018094222102094293573657, 3.31827057369299548889601930734, 5.08193250239214255940378810642, 5.39886040244092591091207033663, 5.92467186361142389098445547120, 7.12954892125448907737267826189, 7.83453299422049901459894856291, 8.903429236009401349480128932011, 10.05673566918767625250193400504
