| L(s) = 1 | + 2·2-s + 3·3-s + 2·4-s + 5-s + 6·6-s − 7-s + 6·9-s + 2·10-s − 3·11-s + 6·12-s − 3·13-s − 2·14-s + 3·15-s − 4·16-s − 2·17-s + 12·18-s + 5·19-s + 2·20-s − 3·21-s − 6·22-s + 25-s − 6·26-s + 9·27-s − 2·28-s − 10·29-s + 6·30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 4-s + 0.447·5-s + 2.44·6-s − 0.377·7-s + 2·9-s + 0.632·10-s − 0.904·11-s + 1.73·12-s − 0.832·13-s − 0.534·14-s + 0.774·15-s − 16-s − 0.485·17-s + 2.82·18-s + 1.14·19-s + 0.447·20-s − 0.654·21-s − 1.27·22-s + 1/5·25-s − 1.17·26-s + 1.73·27-s − 0.377·28-s − 1.85·29-s + 1.09·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.057452055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.057452055\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 - T \) | |
| 151 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−10.05935844887219806968040847569, −9.534769328899119668554728495571, −8.653614521134927480003230121413, −7.63085834570756182140373082846, −6.89828150516592461693418399293, −5.64142856451330350450814329797, −4.74701739835461920500894106990, −3.70239921064209138881299938650, −2.86209258513198245202728340584, −2.20554629541618540322538604956,
2.20554629541618540322538604956, 2.86209258513198245202728340584, 3.70239921064209138881299938650, 4.74701739835461920500894106990, 5.64142856451330350450814329797, 6.89828150516592461693418399293, 7.63085834570756182140373082846, 8.653614521134927480003230121413, 9.534769328899119668554728495571, 10.05935844887219806968040847569
