| L(s) = 1 | − 0.561·2-s + 3·3-s − 7.68·4-s − 6.68·5-s − 1.68·6-s + 8.80·8-s + 9·9-s + 3.75·10-s − 11·11-s − 23.0·12-s − 14.3·13-s − 20.0·15-s + 56.5·16-s + 47.7·17-s − 5.05·18-s − 11.9·19-s + 51.3·20-s + 6.17·22-s − 44.4·23-s + 26.4·24-s − 80.3·25-s + 8.03·26-s + 27·27-s − 139.·29-s + 11.2·30-s + 208.·31-s − 102.·32-s + ⋯ |
| L(s) = 1 | − 0.198·2-s + 0.577·3-s − 0.960·4-s − 0.597·5-s − 0.114·6-s + 0.389·8-s + 0.333·9-s + 0.118·10-s − 0.301·11-s − 0.554·12-s − 0.305·13-s − 0.345·15-s + 0.883·16-s + 0.681·17-s − 0.0661·18-s − 0.143·19-s + 0.574·20-s + 0.0598·22-s − 0.403·23-s + 0.224·24-s − 0.642·25-s + 0.0605·26-s + 0.192·27-s − 0.893·29-s + 0.0685·30-s + 1.20·31-s − 0.564·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.163182073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163182073\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.561T + 8T^{2} \) |
| 5 | \( 1 + 6.68T + 125T^{2} \) |
| 13 | \( 1 + 14.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 139.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 253.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 36.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 114.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 53.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 132.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 817.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 369.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 69.1T + 5.71e5T^{2} \) |
| 89 | \( 1 + 467.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−8.984016298750991871603228847096, −8.073454985151680525321655381733, −7.892410329767637463561555748161, −6.82293675428227086306555591042, −5.60285823965334336744553218352, −4.77134412641342871777728217655, −3.91091384580125958690348202497, −3.20058469469071981814538796976, −1.84411807257265338266219258767, −0.52154588238019340073609344222,
0.52154588238019340073609344222, 1.84411807257265338266219258767, 3.20058469469071981814538796976, 3.91091384580125958690348202497, 4.77134412641342871777728217655, 5.60285823965334336744553218352, 6.82293675428227086306555591042, 7.892410329767637463561555748161, 8.073454985151680525321655381733, 8.984016298750991871603228847096
