| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 14-s + 15-s + 16-s + 6·17-s − 18-s + 5·19-s + 20-s + 21-s + 22-s − 6·23-s − 24-s + 25-s + 27-s + 28-s + 4·29-s − 30-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 − 0.301·11-s + 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 0.742·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.249284727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.249284727\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−14.35230751574960, −13.95298910768502, −13.66257640919461, −12.65124466862347, −12.39706232518948, −11.92076691247148, −11.15498899456849, −10.76368957763065, −10.04471797597923, −9.704397388207638, −9.433268214380358, −8.557852123202844, −8.186566025997427, −7.675950101226009, −7.320364504145090, −6.509543171755220, −5.971040607658442, −5.280445449171553, −4.904135694987455, −3.743604818998865, −3.509699112272800, −2.495966921520573, −2.206036277528913, −1.233580232256722, −0.7526392940932659,
0.7526392940932659, 1.233580232256722, 2.206036277528913, 2.495966921520573, 3.509699112272800, 3.743604818998865, 4.904135694987455, 5.280445449171553, 5.971040607658442, 6.509543171755220, 7.320364504145090, 7.675950101226009, 8.186566025997427, 8.557852123202844, 9.433268214380358, 9.704397388207638, 10.04471797597923, 10.76368957763065, 11.15498899456849, 11.92076691247148, 12.39706232518948, 12.65124466862347, 13.66257640919461, 13.95298910768502, 14.35230751574960
