| L(s) = 1 | + 5.67·3-s + 30.2·7-s + 5.16·9-s − 55.0·11-s − 60.5·13-s + 67.0·17-s − 6.05·19-s + 171.·21-s − 107.·23-s − 123.·27-s − 12.7·29-s + 140.·31-s − 312.·33-s − 149.·37-s − 343.·39-s + 8.36·41-s − 160.·43-s − 133.·47-s + 570.·49-s + 380.·51-s + 150.·53-s − 34.3·57-s + 293.·59-s − 870.·61-s + 156.·63-s − 715.·67-s − 607.·69-s + ⋯ |
| L(s) = 1 | + 1.09·3-s + 1.63·7-s + 0.191·9-s − 1.51·11-s − 1.29·13-s + 0.957·17-s − 0.0731·19-s + 1.78·21-s − 0.970·23-s − 0.882·27-s − 0.0813·29-s + 0.811·31-s − 1.64·33-s − 0.663·37-s − 1.40·39-s + 0.0318·41-s − 0.568·43-s − 0.415·47-s + 1.66·49-s + 1.04·51-s + 0.389·53-s − 0.0798·57-s + 0.647·59-s − 1.82·61-s + 0.312·63-s − 1.30·67-s − 1.05·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 5.67T + 27T^{2} \) |
| 7 | \( 1 - 30.2T + 343T^{2} \) |
| 11 | \( 1 + 55.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.05T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 8.36T + 6.89e4T^{2} \) |
| 43 | \( 1 + 160.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 293.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 870.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 715.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 561.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 46.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 362.T + 9.12e5T^{2} \) |
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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
−8.259900957578221514532463425781, −7.74329204061334893366716259814, −7.39436772279238889976369800663, −5.79720309635097617091978524017, −5.07522766721124986795313585846, −4.40445502277307202966998035201, −3.12688501454163042996327275836, −2.40675350318566947379448764343, −1.61865529839712788626510963014, 0,
1.61865529839712788626510963014, 2.40675350318566947379448764343, 3.12688501454163042996327275836, 4.40445502277307202966998035201, 5.07522766721124986795313585846, 5.79720309635097617091978524017, 7.39436772279238889976369800663, 7.74329204061334893366716259814, 8.259900957578221514532463425781
