| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s − 3·8-s + 9-s + 10-s + 12-s − 13-s − 15-s − 16-s − 2·17-s + 18-s − 4·19-s − 20-s + 8·23-s + 3·24-s + 25-s − 26-s − 27-s + 6·29-s − 30-s + 5·32-s − 2·34-s − 36-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 1.66·23-s + 0.612·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.883·32-s − 0.342·34-s − 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−6.89083098660104736735637779603, −6.65490194830115381999043036199, −5.88304083184638873137066666476, −5.00275853539637462933450778845, −4.88592167258958632098520377881, −3.98526711055699880009319600651, −3.14671228158974323033066489572, −2.33520881512419023373454064611, −1.14128050984593574535963726351, 0,
1.14128050984593574535963726351, 2.33520881512419023373454064611, 3.14671228158974323033066489572, 3.98526711055699880009319600651, 4.88592167258958632098520377881, 5.00275853539637462933450778845, 5.88304083184638873137066666476, 6.65490194830115381999043036199, 6.89083098660104736735637779603
