| L(s) = 1 | + 2-s + 0.347·3-s + 4-s + 0.120·5-s + 0.347·6-s + 1.34·7-s + 8-s − 2.87·9-s + 0.120·10-s + 4.53·11-s + 0.347·12-s + 0.773·13-s + 1.34·14-s + 0.0418·15-s + 16-s + 3·17-s − 2.87·18-s + 2.04·19-s + 0.120·20-s + 0.467·21-s + 4.53·22-s − 0.0564·23-s + 0.347·24-s − 4.98·25-s + 0.773·26-s − 2.04·27-s + 1.34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.200·3-s + 0.5·4-s + 0.0539·5-s + 0.141·6-s + 0.509·7-s + 0.353·8-s − 0.959·9-s + 0.0381·10-s + 1.36·11-s + 0.100·12-s + 0.214·13-s + 0.360·14-s + 0.0108·15-s + 0.250·16-s + 0.727·17-s − 0.678·18-s + 0.468·19-s + 0.0269·20-s + 0.102·21-s + 0.966·22-s − 0.0117·23-s + 0.0708·24-s − 0.997·25-s + 0.151·26-s − 0.392·27-s + 0.254·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.546433757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.546433757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 0.347T + 3T^{2} \) |
| 5 | \( 1 - 0.120T + 5T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 - 0.773T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 + 0.0564T + 23T^{2} \) |
| 29 | \( 1 - 5.78T + 29T^{2} \) |
| 31 | \( 1 - 3.34T + 31T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 9.31T + 43T^{2} \) |
| 47 | \( 1 - 8.51T + 47T^{2} \) |
| 53 | \( 1 - 2.78T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 8.71T + 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 + 7.61T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.551220296356304049555693185845, −8.245547415036518945451241290640, −7.15287075336449639676768472991, −6.43814970690551831226067132234, −5.67028137521222595609967361433, −4.97385731526975260739344103169, −3.93451644304367874412479022902, −3.32172836209824609104664583663, −2.24117667795456605047633570776, −1.13222234732466796548494696636,
1.13222234732466796548494696636, 2.24117667795456605047633570776, 3.32172836209824609104664583663, 3.93451644304367874412479022902, 4.97385731526975260739344103169, 5.67028137521222595609967361433, 6.43814970690551831226067132234, 7.15287075336449639676768472991, 8.245547415036518945451241290640, 8.551220296356304049555693185845
