| L(s) = 1 | + 2·2-s − 3.92·3-s + 4·4-s − 13.4·5-s − 7.84·6-s + 8·8-s − 11.6·9-s − 26.8·10-s − 46.1·11-s − 15.6·12-s + 72.7·13-s + 52.6·15-s + 16·16-s − 51.6·17-s − 23.2·18-s + 115.·19-s − 53.6·20-s − 92.3·22-s + 23·23-s − 31.3·24-s + 54.7·25-s + 145.·26-s + 151.·27-s − 12.8·29-s + 105.·30-s − 41.0·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.755·3-s + 0.5·4-s − 1.19·5-s − 0.533·6-s + 0.353·8-s − 0.429·9-s − 0.847·10-s − 1.26·11-s − 0.377·12-s + 1.55·13-s + 0.905·15-s + 0.250·16-s − 0.736·17-s − 0.303·18-s + 1.39·19-s − 0.599·20-s − 0.894·22-s + 0.208·23-s − 0.266·24-s + 0.437·25-s + 1.09·26-s + 1.07·27-s − 0.0819·29-s + 0.640·30-s − 0.237·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 3.92T + 27T^{2} \) |
| 5 | \( 1 + 13.4T + 125T^{2} \) |
| 11 | \( 1 + 46.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 12.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 41.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 268.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 163.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 259.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 766.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 584.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 713.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 659.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 653.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 91.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 532.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 480.T + 9.12e5T^{2} \) |
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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
−8.128795825342847254635489327642, −7.47884230272032063142866504657, −6.59794437274073031742806092226, −5.74464523215829570369759421311, −5.19167788746508557907479258928, −4.26654256048553818169413110370, −3.46004412349660840496265352374, −2.64580423317126872770516769717, −1.04389384707352477231635530498, 0,
1.04389384707352477231635530498, 2.64580423317126872770516769717, 3.46004412349660840496265352374, 4.26654256048553818169413110370, 5.19167788746508557907479258928, 5.74464523215829570369759421311, 6.59794437274073031742806092226, 7.47884230272032063142866504657, 8.128795825342847254635489327642
