| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 6·11-s + 5·13-s + 14-s + 16-s + 6·17-s − 7·19-s + 20-s − 6·22-s + 6·23-s + 25-s − 5·26-s − 28-s + 3·29-s + 5·31-s − 32-s − 6·34-s − 35-s + 2·37-s + 7·38-s − 40-s + 3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.60·19-s + 0.223·20-s − 1.27·22-s + 1.25·23-s + 1/5·25-s − 0.980·26-s − 0.188·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s − 1.02·34-s − 0.169·35-s + 0.328·37-s + 1.13·38-s − 0.158·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.882259040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882259040\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.640202072953454893453217345019, −7.949761306863157061468193541406, −6.79914965008608226177367078923, −6.40980822725043462065245229339, −5.88732603126508792527143938283, −4.63405673704144098615454524250, −3.69352530542186200833247954335, −2.97279591161022765039706354296, −1.61453512857505661060721008377, −0.990101204987624459755574484830,
0.990101204987624459755574484830, 1.61453512857505661060721008377, 2.97279591161022765039706354296, 3.69352530542186200833247954335, 4.63405673704144098615454524250, 5.88732603126508792527143938283, 6.40980822725043462065245229339, 6.79914965008608226177367078923, 7.949761306863157061468193541406, 8.640202072953454893453217345019
