| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 4·11-s + 12-s − 15-s + 16-s + 2·17-s − 18-s − 4·19-s − 20-s − 4·22-s − 8·23-s − 24-s + 25-s + 27-s + 6·29-s + 30-s + 31-s − 32-s + 4·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-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 − 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.179·31-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 157170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 157170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.55956990961548, −13.05865745107586, −12.35908625163690, −12.12112369250545, −11.59191029047487, −11.25552090152882, −10.45753188283321, −10.12193049018827, −9.703701805637001, −9.219464861516388, −8.495836798355827, −8.366662144291092, −7.930250881078916, −7.297008692297754, −6.636123521213289, −6.456696322515685, −5.854963741387402, −4.972208403217568, −4.445924534752688, −3.842046647911888, −3.431447728665649, −2.802623890609524, −1.984518137973626, −1.625736190646772, −0.8296082715046840, 0,
0.8296082715046840, 1.625736190646772, 1.984518137973626, 2.802623890609524, 3.431447728665649, 3.842046647911888, 4.445924534752688, 4.972208403217568, 5.854963741387402, 6.456696322515685, 6.636123521213289, 7.297008692297754, 7.930250881078916, 8.366662144291092, 8.495836798355827, 9.219464861516388, 9.703701805637001, 10.12193049018827, 10.45753188283321, 11.25552090152882, 11.59191029047487, 12.12112369250545, 12.35908625163690, 13.05865745107586, 13.55956990961548
