L(s) = 1 | + 1.68·2-s + 3·3-s − 5.17·4-s − 9.78·5-s + 5.04·6-s + 6.87·7-s − 22.1·8-s + 9·9-s − 16.4·10-s − 37.8·11-s − 15.5·12-s − 11.3·13-s + 11.5·14-s − 29.3·15-s + 4.12·16-s − 60.3·17-s + 15.1·18-s − 92.5·19-s + 50.6·20-s + 20.6·21-s − 63.6·22-s − 183.·23-s − 66.4·24-s − 29.2·25-s − 19.0·26-s + 27·27-s − 35.5·28-s + ⋯ |
L(s) = 1 | + 0.594·2-s + 0.577·3-s − 0.646·4-s − 0.875·5-s + 0.343·6-s + 0.371·7-s − 0.978·8-s + 0.333·9-s − 0.520·10-s − 1.03·11-s − 0.373·12-s − 0.241·13-s + 0.220·14-s − 0.505·15-s + 0.0643·16-s − 0.860·17-s + 0.198·18-s − 1.11·19-s + 0.565·20-s + 0.214·21-s − 0.616·22-s − 1.66·23-s − 0.565·24-s − 0.233·25-s − 0.143·26-s + 0.192·27-s − 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 + 59T \) |
good | 2 | \( 1 - 1.68T + 8T^{2} \) |
| 5 | \( 1 + 9.78T + 125T^{2} \) |
| 7 | \( 1 - 6.87T + 343T^{2} \) |
| 11 | \( 1 + 37.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 92.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 208.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 393.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 650.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 22.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 209.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 209.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 865.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 843.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 845.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 974.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 338.T + 9.12e5T^{2} \) |
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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
−12.06598782932537296473686986944, −10.79571650174012353228420527526, −9.676630822271598603371694445488, −8.345986232648966630955518299233, −7.974963296889829424368042041371, −6.30817627965155909779523464869, −4.75827180136345138551470840975, −4.05913429019322440350374346581, −2.59330988709362911747075112847, 0,
2.59330988709362911747075112847, 4.05913429019322440350374346581, 4.75827180136345138551470840975, 6.30817627965155909779523464869, 7.974963296889829424368042041371, 8.345986232648966630955518299233, 9.676630822271598603371694445488, 10.79571650174012353228420527526, 12.06598782932537296473686986944