L(s) = 1 | − 1.37·2-s − 3·3-s − 6.10·4-s + 4.12·6-s − 7·7-s + 19.4·8-s + 9·9-s + 15.2·11-s + 18.3·12-s − 76.7·13-s + 9.63·14-s + 22.1·16-s + 96.7·17-s − 12.3·18-s − 14.1·19-s + 21·21-s − 21.0·22-s + 75.7·23-s − 58.2·24-s + 105.·26-s − 27·27-s + 42.7·28-s + 89.9·29-s + 289.·31-s − 185.·32-s − 45.8·33-s − 133.·34-s + ⋯ |
L(s) = 1 | − 0.486·2-s − 0.577·3-s − 0.763·4-s + 0.280·6-s − 0.377·7-s + 0.857·8-s + 0.333·9-s + 0.418·11-s + 0.440·12-s − 1.63·13-s + 0.183·14-s + 0.345·16-s + 1.37·17-s − 0.162·18-s − 0.171·19-s + 0.218·21-s − 0.203·22-s + 0.686·23-s − 0.495·24-s + 0.797·26-s − 0.192·27-s + 0.288·28-s + 0.576·29-s + 1.67·31-s − 1.02·32-s − 0.241·33-s − 0.671·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
good | 2 | \( 1 + 1.37T + 8T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 76.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 96.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 14.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 389.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 228.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 679.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 398.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 891.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 416.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 814.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 650.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.58e3T + 9.12e5T^{2} \) |
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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
−9.858443165779057590530573161390, −9.421124746381115263852894746076, −8.194217428809324361456463219392, −7.41194028829944114252242778900, −6.35983015136586510353046705471, −5.15213361221366533394164205798, −4.47686612313386840621967119383, −3.04340975479234846257269107126, −1.21528244627024320896778762966, 0,
1.21528244627024320896778762966, 3.04340975479234846257269107126, 4.47686612313386840621967119383, 5.15213361221366533394164205798, 6.35983015136586510353046705471, 7.41194028829944114252242778900, 8.194217428809324361456463219392, 9.421124746381115263852894746076, 9.858443165779057590530573161390