| L(s) = 1 | − 2-s + 4-s + 2.78·5-s + 2.18·7-s − 8-s − 2.78·10-s − 3.79·11-s + 3.92·13-s − 2.18·14-s + 16-s − 7.46·17-s − 5.81·19-s + 2.78·20-s + 3.79·22-s − 3.28·23-s + 2.75·25-s − 3.92·26-s + 2.18·28-s − 5.39·29-s + 1.61·31-s − 32-s + 7.46·34-s + 6.08·35-s + 4.51·37-s + 5.81·38-s − 2.78·40-s + 2.46·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.24·5-s + 0.825·7-s − 0.353·8-s − 0.880·10-s − 1.14·11-s + 1.08·13-s − 0.583·14-s + 0.250·16-s − 1.81·17-s − 1.33·19-s + 0.622·20-s + 0.808·22-s − 0.684·23-s + 0.550·25-s − 0.770·26-s + 0.412·28-s − 1.00·29-s + 0.289·31-s − 0.176·32-s + 1.28·34-s + 1.02·35-s + 0.741·37-s + 0.943·38-s − 0.440·40-s + 0.384·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 + 5.81T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 31 | \( 1 - 1.61T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 6.51T + 43T^{2} \) |
| 47 | \( 1 - 9.17T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 5.07T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 4.21T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 4.27T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 97 | \( 1 - 1.13T + 97T^{2} \) |
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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
−7.70214375554557356571518689989, −6.74497202189527825609325387343, −6.04969008429768270830651175836, −5.70996470370743144672045154211, −4.67694510017763983522701520916, −4.01944785606618707203815986338, −2.55988982401000849370830540684, −2.18744887627079361139361592394, −1.41106865631353533920862965922, 0,
1.41106865631353533920862965922, 2.18744887627079361139361592394, 2.55988982401000849370830540684, 4.01944785606618707203815986338, 4.67694510017763983522701520916, 5.70996470370743144672045154211, 6.04969008429768270830651175836, 6.74497202189527825609325387343, 7.70214375554557356571518689989
