| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 4·7-s + 9-s − 2·10-s − 3·11-s − 2·12-s + 5·13-s + 8·14-s + 15-s − 4·16-s − 7·17-s + 2·18-s − 2·20-s − 4·21-s − 6·22-s − 9·23-s + 25-s + 10·26-s − 27-s + 8·28-s + 2·30-s + 7·31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 1.51·7-s + 1/3·9-s − 0.632·10-s − 0.904·11-s − 0.577·12-s + 1.38·13-s + 2.13·14-s + 0.258·15-s − 16-s − 1.69·17-s + 0.471·18-s − 0.447·20-s − 0.872·21-s − 1.27·22-s − 1.87·23-s + 1/5·25-s + 1.96·26-s − 0.192·27-s + 1.51·28-s + 0.365·30-s + 1.25·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.433689181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.433689181\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−15.34503753691967, −14.62323697944602, −14.04385467337440, −13.67835383273171, −13.12391481621238, −12.71622437637488, −11.93979834422841, −11.49218517262294, −11.25027594184443, −10.73933316411502, −10.11092933699955, −9.113528271707296, −8.469331478183816, −8.004992714357532, −7.502435524301712, −6.410041324612984, −6.248686243490987, −5.583048973426164, −4.775728317858799, −4.526156166044958, −4.073067294847146, −3.244365100888598, −2.311870338037015, −1.776223142260624, −0.5762841000079583,
0.5762841000079583, 1.776223142260624, 2.311870338037015, 3.244365100888598, 4.073067294847146, 4.526156166044958, 4.775728317858799, 5.583048973426164, 6.248686243490987, 6.410041324612984, 7.502435524301712, 8.004992714357532, 8.469331478183816, 9.113528271707296, 10.11092933699955, 10.73933316411502, 11.25027594184443, 11.49218517262294, 11.93979834422841, 12.71622437637488, 13.12391481621238, 13.67835383273171, 14.04385467337440, 14.62323697944602, 15.34503753691967
