| L(s) = 1 | − 4·2-s − 21.7·3-s + 16·4-s + 87.1·6-s + 49·7-s − 64·8-s + 231.·9-s − 56.3·11-s − 348.·12-s − 612.·13-s − 196·14-s + 256·16-s − 358.·17-s − 924.·18-s + 925.·19-s − 1.06e3·21-s + 225.·22-s − 2.15e3·23-s + 1.39e3·24-s + 2.45e3·26-s + 256.·27-s + 784·28-s + 6.33e3·29-s + 2.29e3·31-s − 1.02e3·32-s + 1.22e3·33-s + 1.43e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.987·6-s + 0.377·7-s − 0.353·8-s + 0.951·9-s − 0.140·11-s − 0.698·12-s − 1.00·13-s − 0.267·14-s + 0.250·16-s − 0.300·17-s − 0.672·18-s + 0.588·19-s − 0.527·21-s + 0.0992·22-s − 0.850·23-s + 0.493·24-s + 0.710·26-s + 0.0677·27-s + 0.188·28-s + 1.39·29-s + 0.428·31-s − 0.176·32-s + 0.196·33-s + 0.212·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 21.7T + 243T^{2} \) |
| 11 | \( 1 + 56.3T + 1.61e5T^{2} \) |
| 13 | \( 1 + 612.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 358.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 925.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 835.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.81e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.13e5T + 8.58e9T^{2} \) |
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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
−10.33711526897830866645047702770, −9.497879106460474700906934208241, −8.243669381931985321088628439167, −7.28620597237907357541720528576, −6.34820271868283375185870671960, −5.40681831547882218165023702127, −4.44678942731446812111127286805, −2.55633352028904637032941960992, −1.06705904140191035540692314601, 0,
1.06705904140191035540692314601, 2.55633352028904637032941960992, 4.44678942731446812111127286805, 5.40681831547882218165023702127, 6.34820271868283375185870671960, 7.28620597237907357541720528576, 8.243669381931985321088628439167, 9.497879106460474700906934208241, 10.33711526897830866645047702770
