| L(s) = 1 | − 2.73·2-s + 5.46·4-s + 3.73·5-s − 7-s − 9.46·8-s − 10.1·10-s − 0.732·13-s + 2.73·14-s + 14.9·16-s + 0.267·17-s + 8.19·19-s + 20.3·20-s − 6.73·23-s + 8.92·25-s + 2·26-s − 5.46·28-s − 4.73·29-s + 0.535·31-s − 21.8·32-s − 0.732·34-s − 3.73·35-s − 2.53·37-s − 22.3·38-s − 35.3·40-s − 4·41-s − 6.46·43-s + 18.3·46-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.73·4-s + 1.66·5-s − 0.377·7-s − 3.34·8-s − 3.22·10-s − 0.203·13-s + 0.730·14-s + 3.73·16-s + 0.0649·17-s + 1.88·19-s + 4.55·20-s − 1.40·23-s + 1.78·25-s + 0.392·26-s − 1.03·28-s − 0.878·29-s + 0.0962·31-s − 3.86·32-s − 0.125·34-s − 0.630·35-s − 0.416·37-s − 3.63·38-s − 5.58·40-s − 0.624·41-s − 0.985·43-s + 2.71·46-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 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 13 | \( 1 + 0.732T + 13T^{2} \) |
| 17 | \( 1 - 0.267T + 17T^{2} \) |
| 19 | \( 1 - 8.19T + 19T^{2} \) |
| 23 | \( 1 + 6.73T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 + 2.53T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 3.26T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 + 7.73T + 83T^{2} \) |
| 89 | \( 1 + 2.66T + 89T^{2} \) |
| 97 | \( 1 - 6.73T + 97T^{2} \) |
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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
−7.63904652206657779081458307581, −7.00975015173377592437661304972, −6.28061646112836672212387731178, −5.83302423305163685350634578220, −5.10373080308573760173316594488, −3.39543698488040164789938736829, −2.72284744247203233236987706637, −1.80965353932400059676799728791, −1.36079094486045536322818369327, 0,
1.36079094486045536322818369327, 1.80965353932400059676799728791, 2.72284744247203233236987706637, 3.39543698488040164789938736829, 5.10373080308573760173316594488, 5.83302423305163685350634578220, 6.28061646112836672212387731178, 7.00975015173377592437661304972, 7.63904652206657779081458307581
