| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s + 12-s + 13-s − 14-s + 16-s − 4·17-s − 18-s + 21-s − 24-s − 5·25-s − 26-s + 27-s + 28-s − 2·29-s − 2·31-s − 32-s + 4·34-s + 36-s − 8·37-s + 39-s + 2·41-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 + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.218·21-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 1/6·36-s − 1.31·37-s + 0.160·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.550772490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550772490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.20525840707104, −13.63763811501331, −13.38645401558715, −12.65602840758963, −12.14674073285936, −11.57126066934437, −11.12061054391687, −10.59399282981814, −10.11619629517154, −9.519850410455866, −9.002655798112919, −8.644278911652728, −8.091254444094846, −7.589423330432867, −7.069048292077101, −6.526089857409628, −5.901682429970419, −5.239205966466950, −4.586851775377504, −3.845767798037626, −3.418272891384533, −2.520888981092585, −1.991815144586546, −1.460804831732404, −0.4410115967940642,
0.4410115967940642, 1.460804831732404, 1.991815144586546, 2.520888981092585, 3.418272891384533, 3.845767798037626, 4.586851775377504, 5.239205966466950, 5.901682429970419, 6.526089857409628, 7.069048292077101, 7.589423330432867, 8.091254444094846, 8.644278911652728, 9.002655798112919, 9.519850410455866, 10.11619629517154, 10.59399282981814, 11.12061054391687, 11.57126066934437, 12.14674073285936, 12.65602840758963, 13.38645401558715, 13.63763811501331, 14.20525840707104
