| L(s) = 1 | − 0.347·2-s − 1.87·4-s − 2.41·5-s − 2.87·7-s + 1.34·8-s + 0.837·10-s + 11-s − 5.71·13-s + 14-s + 3.29·16-s − 3.65·17-s + 6.57·19-s + 4.53·20-s − 0.347·22-s + 3.75·23-s + 0.815·25-s + 1.98·26-s + 5.41·28-s − 7.14·29-s − 3.45·31-s − 3.83·32-s + 1.26·34-s + 6.94·35-s + 6.82·37-s − 2.28·38-s − 3.24·40-s + 4.98·41-s + ⋯ |
| L(s) = 1 | − 0.245·2-s − 0.939·4-s − 1.07·5-s − 1.08·7-s + 0.476·8-s + 0.264·10-s + 0.301·11-s − 1.58·13-s + 0.267·14-s + 0.822·16-s − 0.885·17-s + 1.50·19-s + 1.01·20-s − 0.0740·22-s + 0.783·23-s + 0.163·25-s + 0.389·26-s + 1.02·28-s − 1.32·29-s − 0.620·31-s − 0.678·32-s + 0.217·34-s + 1.17·35-s + 1.12·37-s − 0.370·38-s − 0.513·40-s + 0.778·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.347T + 2T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 6.57T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 + 7.14T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 4.98T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 + 7.57T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 + 5.06T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 - 2.04T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 8.83T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.47247065966688644908133265944, −7.13020553302642157665424707022, −6.12133314793847534713871989460, −5.20452947472943603314683649267, −4.66207716718087692689957474106, −3.81036240946113370588422172190, −3.35021201332195858573244892021, −2.33019044350758157966035914230, −0.797967726415448087895937809101, 0,
0.797967726415448087895937809101, 2.33019044350758157966035914230, 3.35021201332195858573244892021, 3.81036240946113370588422172190, 4.66207716718087692689957474106, 5.20452947472943603314683649267, 6.12133314793847534713871989460, 7.13020553302642157665424707022, 7.47247065966688644908133265944
