| L(s) = 1 | − 1.36·2-s − 0.141·4-s − 3.14·5-s + 7-s + 2.91·8-s + 4.28·10-s + 4.77·13-s − 1.36·14-s − 3.69·16-s − 4.77·17-s + 7.00·19-s + 0.443·20-s − 5.14·23-s + 4.86·25-s − 6.51·26-s − 0.141·28-s + 7.00·29-s − 3.63·31-s − 0.797·32-s + 6.51·34-s − 3.14·35-s − 9.86·37-s − 9.55·38-s − 9.17·40-s − 3.22·41-s − 4.28·43-s + 7.00·46-s + ⋯ |
| L(s) = 1 | − 0.964·2-s − 0.0706·4-s − 1.40·5-s + 0.377·7-s + 1.03·8-s + 1.35·10-s + 1.32·13-s − 0.364·14-s − 0.924·16-s − 1.15·17-s + 1.60·19-s + 0.0992·20-s − 1.07·23-s + 0.973·25-s − 1.27·26-s − 0.0267·28-s + 1.30·29-s − 0.653·31-s − 0.141·32-s + 1.11·34-s − 0.530·35-s − 1.62·37-s − 1.55·38-s − 1.45·40-s − 0.503·41-s − 0.653·43-s + 1.03·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 - 7.00T + 19T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 + 3.63T + 31T^{2} \) |
| 37 | \( 1 + 9.86T + 37T^{2} \) |
| 41 | \( 1 + 3.22T + 41T^{2} \) |
| 43 | \( 1 + 4.28T + 43T^{2} \) |
| 47 | \( 1 - 0.778T + 47T^{2} \) |
| 53 | \( 1 + 2.28T + 53T^{2} \) |
| 59 | \( 1 - 0.363T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 + 6.59T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 - 1.55T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 - 6.15T + 97T^{2} \) |
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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
−7.74964893516527937149628308276, −7.12058810388824924021724994207, −6.41687336283557190778323553272, −5.27710038355053337136107573841, −4.61652567720975334695143655297, −3.86055606185165122179937970553, −3.33183148283014022041928849014, −1.90172089076921706656830620604, −0.991645112262633184237523442496, 0,
0.991645112262633184237523442496, 1.90172089076921706656830620604, 3.33183148283014022041928849014, 3.86055606185165122179937970553, 4.61652567720975334695143655297, 5.27710038355053337136107573841, 6.41687336283557190778323553272, 7.12058810388824924021724994207, 7.74964893516527937149628308276
