| L(s) = 1 | + 2-s − 2.53·3-s + 4-s − 0.652·5-s − 2.53·6-s + 7-s + 8-s + 3.41·9-s − 0.652·10-s + 0.532·11-s − 2.53·12-s + 1.87·13-s + 14-s + 1.65·15-s + 16-s − 1.04·17-s + 3.41·18-s − 0.652·20-s − 2.53·21-s + 0.532·22-s − 8.82·23-s − 2.53·24-s − 4.57·25-s + 1.87·26-s − 1.04·27-s + 28-s − 4.81·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.46·3-s + 0.5·4-s − 0.291·5-s − 1.03·6-s + 0.377·7-s + 0.353·8-s + 1.13·9-s − 0.206·10-s + 0.160·11-s − 0.730·12-s + 0.521·13-s + 0.267·14-s + 0.426·15-s + 0.250·16-s − 0.252·17-s + 0.804·18-s − 0.145·20-s − 0.552·21-s + 0.113·22-s − 1.83·23-s − 0.516·24-s − 0.914·25-s + 0.368·26-s − 0.200·27-s + 0.188·28-s − 0.894·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.53T + 3T^{2} \) |
| 5 | \( 1 + 0.652T + 5T^{2} \) |
| 11 | \( 1 - 0.532T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 + 1.04T + 17T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 4.81T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 + 7.36T + 43T^{2} \) |
| 47 | \( 1 + 0.411T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 - 2.73T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 - 3.43T + 67T^{2} \) |
| 71 | \( 1 + 0.554T + 71T^{2} \) |
| 73 | \( 1 - 0.170T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 5.38T + 97T^{2} \) |
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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
−7.77561965244429940889764076900, −6.87312884115100455006396356450, −6.10754534088389438249603745371, −5.84721320615097025892865316409, −4.96048759842280866786746061566, −4.27394665307009153602069525368, −3.68487258486702079381383305646, −2.33506187518766204656078071439, −1.29897141774553520352100295367, 0,
1.29897141774553520352100295367, 2.33506187518766204656078071439, 3.68487258486702079381383305646, 4.27394665307009153602069525368, 4.96048759842280866786746061566, 5.84721320615097025892865316409, 6.10754534088389438249603745371, 6.87312884115100455006396356450, 7.77561965244429940889764076900
