| L(s) = 1 | + 0.305·2-s − 1.90·4-s + 5-s + 1.72·7-s − 1.19·8-s + 0.305·10-s − 5.65·13-s + 0.527·14-s + 3.44·16-s − 4.85·17-s + 2·19-s − 1.90·20-s + 2.20·23-s + 25-s − 1.72·26-s − 3.29·28-s − 4.24·29-s − 1.45·31-s + 3.43·32-s − 1.48·34-s + 1.72·35-s + 6.85·37-s + 0.610·38-s − 1.19·40-s + 12.1·41-s + 4.03·43-s + 0.674·46-s + ⋯ |
| L(s) = 1 | + 0.215·2-s − 0.953·4-s + 0.447·5-s + 0.652·7-s − 0.421·8-s + 0.0965·10-s − 1.56·13-s + 0.140·14-s + 0.862·16-s − 1.17·17-s + 0.458·19-s − 0.426·20-s + 0.460·23-s + 0.200·25-s − 0.338·26-s − 0.622·28-s − 0.788·29-s − 0.261·31-s + 0.607·32-s − 0.254·34-s + 0.291·35-s + 1.12·37-s + 0.0990·38-s − 0.188·40-s + 1.90·41-s + 0.615·43-s + 0.0994·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.582272312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582272312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 0.305T + 2T^{2} \) |
| 7 | \( 1 - 1.72T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.20T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 1.45T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 - 6.69T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 - 2.63T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 + 7.79T + 79T^{2} \) |
| 83 | \( 1 + 1.24T + 83T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.542518824008484233232747799524, −7.65873381142234381812392601302, −7.15499070701468981496011391220, −5.99024072736837976796211400463, −5.38261231589774788813003948182, −4.61744974597789883669535663863, −4.17540197329886521415262795370, −2.88207887628285819460904448663, −2.08200325769082879246528953886, −0.69490746642242851662255696656,
0.69490746642242851662255696656, 2.08200325769082879246528953886, 2.88207887628285819460904448663, 4.17540197329886521415262795370, 4.61744974597789883669535663863, 5.38261231589774788813003948182, 5.99024072736837976796211400463, 7.15499070701468981496011391220, 7.65873381142234381812392601302, 8.542518824008484233232747799524
