| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 18-s − 4·19-s − 20-s + 21-s − 22-s − 9·23-s + 24-s + 25-s − 26-s − 27-s − 28-s + 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s − 1.87·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 31 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.65579663707050, −12.42764468927262, −11.77489684086895, −11.43902574157601, −11.18964943810947, −10.38862014470112, −10.16271075492046, −9.856673730280586, −9.191064015151587, −8.623560245739180, −8.354380398524998, −7.799922870289043, −7.378141193887496, −6.696454127657659, −6.344357276376963, −6.123720622172859, −5.383181203498756, −4.786104493647361, −4.298290253848016, −3.609536006658240, −3.390448834567858, −2.425176513030937, −1.958682025037782, −1.347883220271240, −0.5440000949866078, 0,
0.5440000949866078, 1.347883220271240, 1.958682025037782, 2.425176513030937, 3.390448834567858, 3.609536006658240, 4.298290253848016, 4.786104493647361, 5.383181203498756, 6.123720622172859, 6.344357276376963, 6.696454127657659, 7.378141193887496, 7.799922870289043, 8.354380398524998, 8.623560245739180, 9.191064015151587, 9.856673730280586, 10.16271075492046, 10.38862014470112, 11.18964943810947, 11.43902574157601, 11.77489684086895, 12.42764468927262, 12.65579663707050
