| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 4·7-s − 8-s + 9-s − 3·10-s + 3·11-s + 12-s − 13-s + 4·14-s + 3·15-s + 16-s − 17-s − 18-s − 4·19-s + 3·20-s − 4·21-s − 3·22-s − 3·23-s − 24-s + 4·25-s + 26-s + 27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.670·20-s − 0.872·21-s − 0.639·22-s − 0.625·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225318 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225318 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| 47 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.08243195888286, −12.86565950768374, −12.38116779691322, −11.80444823448091, −11.19220133383787, −10.65187940074711, −10.11621242265518, −9.775192606322898, −9.531267613285919, −9.087547338502619, −8.610755731138530, −8.254969256746233, −7.281649926689709, −7.041670894790376, −6.483332371926145, −6.243850468152069, −5.569366003763270, −5.212520497153088, −4.084831034422603, −3.806429058647795, −3.223012657511990, −2.491693723465332, −2.066415707634348, −1.699959026551390, −0.7710558057863530, 0,
0.7710558057863530, 1.699959026551390, 2.066415707634348, 2.491693723465332, 3.223012657511990, 3.806429058647795, 4.084831034422603, 5.212520497153088, 5.569366003763270, 6.243850468152069, 6.483332371926145, 7.041670894790376, 7.281649926689709, 8.254969256746233, 8.610755731138530, 9.087547338502619, 9.531267613285919, 9.775192606322898, 10.11621242265518, 10.65187940074711, 11.19220133383787, 11.80444823448091, 12.38116779691322, 12.86565950768374, 13.08243195888286
