| L(s) = 1 | − 1.36·2-s − 3.15·3-s − 6.14·4-s + 4.29·6-s + 7.94·7-s + 19.2·8-s − 17.0·9-s + 27.6·11-s + 19.3·12-s − 58.1·13-s − 10.8·14-s + 22.9·16-s + 17·17-s + 23.2·18-s + 89.1·19-s − 25.0·21-s − 37.5·22-s + 115.·23-s − 60.7·24-s + 79.1·26-s + 138.·27-s − 48.8·28-s − 128.·29-s + 273.·31-s − 185.·32-s − 87.1·33-s − 23.1·34-s + ⋯ |
| L(s) = 1 | − 0.481·2-s − 0.607·3-s − 0.768·4-s + 0.292·6-s + 0.428·7-s + 0.851·8-s − 0.631·9-s + 0.756·11-s + 0.466·12-s − 1.23·13-s − 0.206·14-s + 0.358·16-s + 0.242·17-s + 0.303·18-s + 1.07·19-s − 0.260·21-s − 0.364·22-s + 1.04·23-s − 0.516·24-s + 0.596·26-s + 0.990·27-s − 0.329·28-s − 0.823·29-s + 1.58·31-s − 1.02·32-s − 0.459·33-s − 0.116·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 1.36T + 8T^{2} \) |
| 3 | \( 1 + 3.15T + 27T^{2} \) |
| 7 | \( 1 - 7.94T + 343T^{2} \) |
| 11 | \( 1 - 27.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.1T + 2.19e3T^{2} \) |
| 19 | \( 1 - 89.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 527.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 53.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 52.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 788.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 295.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 720.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 116.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 813.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 794.T + 9.12e5T^{2} \) |
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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
−10.10857585543989119558132378762, −9.494266827677247065775395817896, −8.533718429210806434571912870325, −7.67981882478959126087242350651, −6.58366669112844742836502536899, −5.23655704781515135014069645141, −4.73807994659988524854829585484, −3.18305995834881609217414097336, −1.30860436236348860180256247830, 0,
1.30860436236348860180256247830, 3.18305995834881609217414097336, 4.73807994659988524854829585484, 5.23655704781515135014069645141, 6.58366669112844742836502536899, 7.67981882478959126087242350651, 8.533718429210806434571912870325, 9.494266827677247065775395817896, 10.10857585543989119558132378762
