| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 2·11-s + 12-s + 13-s − 14-s + 15-s + 16-s − 8·17-s − 18-s + 20-s + 21-s + 2·22-s + 23-s − 24-s − 4·25-s − 26-s + 27-s + 28-s + 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−12.72954066216235, −12.39732693492015, −11.64209619778366, −11.46468835679546, −10.73799083590209, −10.50154484286182, −10.09795808752366, −9.508606504775998, −9.043765611092346, −8.672468793096202, −8.341547637039260, −7.882138574905175, −7.212931011306682, −6.910443570253875, −6.429707955156870, −5.874105220809344, −5.202140500229304, −4.902406984091214, −4.062990665882983, −3.796227470633628, −2.909980244540255, −2.499853004897369, −1.963612378821214, −1.614332999301317, −0.7278354331538418, 0,
0.7278354331538418, 1.614332999301317, 1.963612378821214, 2.499853004897369, 2.909980244540255, 3.796227470633628, 4.062990665882983, 4.902406984091214, 5.202140500229304, 5.874105220809344, 6.429707955156870, 6.910443570253875, 7.212931011306682, 7.882138574905175, 8.341547637039260, 8.672468793096202, 9.043765611092346, 9.508606504775998, 10.09795808752366, 10.50154484286182, 10.73799083590209, 11.46468835679546, 11.64209619778366, 12.39732693492015, 12.72954066216235
