| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s + 6·13-s − 15-s + 16-s + 2·17-s − 18-s − 4·19-s − 20-s − 24-s + 25-s − 6·26-s + 27-s + 10·29-s + 30-s − 32-s − 2·34-s + 36-s + 6·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 1.85·29-s + 0.182·30-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.568162078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.568162078\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.19289581273586, −12.71125411220735, −12.09911157181303, −11.82782521966749, −11.02043650919975, −10.86055648282063, −10.34771106971765, −9.827383468833422, −9.246708277397060, −8.810951106513139, −8.310940855220700, −8.115656937354966, −7.605334729553178, −6.839909754737622, −6.544880416443280, −5.987904406129502, −5.466760950353856, −4.456468471708070, −4.285110523928950, −3.516882048272913, −3.034907611049958, −2.526411937239580, −1.693548991080971, −1.162151391121539, −0.5496751808076520,
0.5496751808076520, 1.162151391121539, 1.693548991080971, 2.526411937239580, 3.034907611049958, 3.516882048272913, 4.285110523928950, 4.456468471708070, 5.466760950353856, 5.987904406129502, 6.544880416443280, 6.839909754737622, 7.605334729553178, 8.115656937354966, 8.310940855220700, 8.810951106513139, 9.246708277397060, 9.827383468833422, 10.34771106971765, 10.86055648282063, 11.02043650919975, 11.82782521966749, 12.09911157181303, 12.71125411220735, 13.19289581273586
