| 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 + 4·19-s + 20-s − 21-s − 22-s − 24-s + 25-s + 27-s − 28-s + 6·29-s − 30-s − 8·31-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 + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.969454590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.969454590\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.46742345716090, −12.00376519027747, −11.57203968792082, −10.89486445210280, −10.62339806521517, −10.02256192651900, −9.612475955644735, −9.388485314675121, −8.968337765924691, −8.394605348702471, −7.856453186023182, −7.476948747113066, −7.202002076561601, −6.491843908050262, −5.950506159238874, −5.714181599896481, −5.039442205915302, −4.395375554977274, −3.783859030906239, −3.268410353466391, −2.807328930769400, −2.336152532903145, −1.617866254897141, −1.024493813220823, −0.6421024853064671,
0.6421024853064671, 1.024493813220823, 1.617866254897141, 2.336152532903145, 2.807328930769400, 3.268410353466391, 3.783859030906239, 4.395375554977274, 5.039442205915302, 5.714181599896481, 5.950506159238874, 6.491843908050262, 7.202002076561601, 7.476948747113066, 7.856453186023182, 8.394605348702471, 8.968337765924691, 9.388485314675121, 9.612475955644735, 10.02256192651900, 10.62339806521517, 10.89486445210280, 11.57203968792082, 12.00376519027747, 12.46742345716090
