| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 6·11-s + 2·15-s + 2·17-s + 4·21-s − 8·23-s − 25-s − 27-s + 2·29-s − 8·31-s − 6·33-s + 8·35-s + 8·37-s + 2·41-s − 8·43-s − 2·45-s + 6·47-s + 9·49-s − 2·51-s + 6·53-s − 12·55-s + 2·59-s + 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.80·11-s + 0.516·15-s + 0.485·17-s + 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 1.04·33-s + 1.35·35-s + 1.31·37-s + 0.312·41-s − 1.21·43-s − 0.298·45-s + 0.875·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 1.61·55-s + 0.260·59-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.34827461469256113909646088957, −6.68785007511888502346458446160, −6.19243636616154315461632127526, −5.62603373067719656010963818745, −4.43149074002984843123938895814, −3.71419411709342416641494259611, −3.56945303063676622727099002736, −2.19962452120547486932148396908, −0.977961936386158244281216620916, 0,
0.977961936386158244281216620916, 2.19962452120547486932148396908, 3.56945303063676622727099002736, 3.71419411709342416641494259611, 4.43149074002984843123938895814, 5.62603373067719656010963818745, 6.19243636616154315461632127526, 6.68785007511888502346458446160, 7.34827461469256113909646088957
