| L(s) = 1 | − 2-s + 4-s + 1.10·5-s − 7-s − 8-s − 1.10·10-s + 5.62·11-s − 3.62·13-s + 14-s + 16-s − 4.52·17-s − 0.578·19-s + 1.10·20-s − 5.62·22-s + 23-s − 3.78·25-s + 3.62·26-s − 28-s − 5.83·29-s − 2.52·31-s − 32-s + 4.52·34-s − 1.10·35-s − 7.04·37-s + 0.578·38-s − 1.10·40-s − 3.15·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.493·5-s − 0.377·7-s − 0.353·8-s − 0.348·10-s + 1.69·11-s − 1.00·13-s + 0.267·14-s + 0.250·16-s − 1.09·17-s − 0.132·19-s + 0.246·20-s − 1.19·22-s + 0.208·23-s − 0.756·25-s + 0.711·26-s − 0.188·28-s − 1.08·29-s − 0.453·31-s − 0.176·32-s + 0.775·34-s − 0.186·35-s − 1.15·37-s + 0.0938·38-s − 0.174·40-s − 0.492·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 1.10T + 5T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 0.578T + 19T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 + 3.15T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 6.72T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.688421445177832054212307880552, −7.54052370674942231527243705140, −6.90967017242737000618647967810, −6.32412875592042846101025341642, −5.49214223325756869520848848095, −4.36812667418581342929479818581, −3.51621945116073028453100034819, −2.31012704792875966104294639173, −1.54088432072019553065177892743, 0,
1.54088432072019553065177892743, 2.31012704792875966104294639173, 3.51621945116073028453100034819, 4.36812667418581342929479818581, 5.49214223325756869520848848095, 6.32412875592042846101025341642, 6.90967017242737000618647967810, 7.54052370674942231527243705140, 8.688421445177832054212307880552
