| L(s) = 1 | + 2.31·2-s − 2.63·4-s + 5·5-s − 24.6·8-s + 11.5·10-s − 46.2·11-s + 61.3·13-s − 36·16-s + 101.·17-s − 3.66·19-s − 13.1·20-s − 107.·22-s − 84.8·23-s + 25·25-s + 142.·26-s − 30.1·29-s − 188.·31-s + 113.·32-s + 234.·34-s + 18.0·37-s − 8.49·38-s − 123.·40-s + 481.·41-s − 97.7·43-s + 121.·44-s − 196.·46-s + 117.·47-s + ⋯ |
| L(s) = 1 | + 0.819·2-s − 0.329·4-s + 0.447·5-s − 1.08·8-s + 0.366·10-s − 1.26·11-s + 1.30·13-s − 0.562·16-s + 1.44·17-s − 0.0442·19-s − 0.147·20-s − 1.03·22-s − 0.769·23-s + 0.200·25-s + 1.07·26-s − 0.193·29-s − 1.09·31-s + 0.627·32-s + 1.18·34-s + 0.0802·37-s − 0.0362·38-s − 0.486·40-s + 1.83·41-s − 0.346·43-s + 0.417·44-s − 0.630·46-s + 0.365·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.31T + 8T^{2} \) |
| 11 | \( 1 + 46.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 3.66T + 6.85e3T^{2} \) |
| 23 | \( 1 + 84.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 30.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 481.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 667.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 57.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 552.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 740.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 233.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 683.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 218.T + 9.12e5T^{2} \) |
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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
−8.202626515721192135736894210948, −7.68389165186977111770916225327, −6.39496995939240137948591914470, −5.68183981694589348740438607502, −5.31536806417530739180749664975, −4.22101127843275003268718989771, −3.44822034718134925140557091536, −2.62638037376683885057015890499, −1.30568084041543222898437621453, 0,
1.30568084041543222898437621453, 2.62638037376683885057015890499, 3.44822034718134925140557091536, 4.22101127843275003268718989771, 5.31536806417530739180749664975, 5.68183981694589348740438607502, 6.39496995939240137948591914470, 7.68389165186977111770916225327, 8.202626515721192135736894210948
