| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 3·11-s − 12-s − 14-s + 16-s − 17-s − 18-s + 8·19-s − 21-s − 3·22-s + 4·23-s + 24-s − 27-s + 28-s − 7·29-s − 31-s − 32-s − 3·33-s + 34-s + 36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.218·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 0.192·27-s + 0.188·28-s − 1.29·29-s − 0.179·31-s − 0.176·32-s − 0.522·33-s + 0.171·34-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−15.79128567292302, −15.09453618190712, −14.67188648963622, −14.13011204480372, −13.36231392284636, −12.94041494030004, −12.14001834615935, −11.62849345420527, −11.35678635083201, −10.87071867102931, −9.997463134694294, −9.681781341194593, −9.092128257861596, −8.521782588398079, −7.833472030783972, −7.152559819796156, −6.877842552376929, −6.119733746426954, −5.307121242485032, −5.081163119127009, −3.975508923983301, −3.461673604110862, −2.549640631458183, −1.531281759800074, −1.130965165453461, 0,
1.130965165453461, 1.531281759800074, 2.549640631458183, 3.461673604110862, 3.975508923983301, 5.081163119127009, 5.307121242485032, 6.119733746426954, 6.877842552376929, 7.152559819796156, 7.833472030783972, 8.521782588398079, 9.092128257861596, 9.681781341194593, 9.997463134694294, 10.87071867102931, 11.35678635083201, 11.62849345420527, 12.14001834615935, 12.94041494030004, 13.36231392284636, 14.13011204480372, 14.67188648963622, 15.09453618190712, 15.79128567292302
