| L(s) = 1 | + 0.816·2-s − 3·3-s − 7.33·4-s − 2.99·5-s − 2.44·6-s − 12.5·8-s + 9·9-s − 2.44·10-s + 11·11-s + 21.9·12-s + 41.2·13-s + 8.99·15-s + 48.4·16-s − 59.9·17-s + 7.34·18-s + 16.0·19-s + 21.9·20-s + 8.98·22-s + 52.1·23-s + 37.5·24-s − 116.·25-s + 33.6·26-s − 27·27-s − 124.·29-s + 7.34·30-s − 90.4·31-s + 139.·32-s + ⋯ |
| L(s) = 1 | + 0.288·2-s − 0.577·3-s − 0.916·4-s − 0.268·5-s − 0.166·6-s − 0.553·8-s + 0.333·9-s − 0.0774·10-s + 0.301·11-s + 0.529·12-s + 0.879·13-s + 0.154·15-s + 0.756·16-s − 0.854·17-s + 0.0962·18-s + 0.193·19-s + 0.245·20-s + 0.0870·22-s + 0.472·23-s + 0.319·24-s − 0.928·25-s + 0.254·26-s − 0.192·27-s − 0.796·29-s + 0.0447·30-s − 0.523·31-s + 0.771·32-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
| \(L(2)\) |
\(\approx\) |
\(0.9835282422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9835282422\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.816T + 8T^{2} \) |
| 5 | \( 1 + 2.99T + 125T^{2} \) |
| 13 | \( 1 - 41.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 285.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 226.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 526.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 886.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 76.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 402.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 582.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 32.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 772.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 576.T + 9.12e5T^{2} \) |
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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
−9.033715685419241674536975332396, −8.365422628815417415165601625724, −7.39578237470038305108681812419, −6.44336652291156884706954400416, −5.68002220782371555048126458876, −4.89470906934542057969518815942, −4.02706849919695224031626635233, −3.37870518445278895424408886783, −1.73064745875571742144252475370, −0.48018816597494195639222598787,
0.48018816597494195639222598787, 1.73064745875571742144252475370, 3.37870518445278895424408886783, 4.02706849919695224031626635233, 4.89470906934542057969518815942, 5.68002220782371555048126458876, 6.44336652291156884706954400416, 7.39578237470038305108681812419, 8.365422628815417415165601625724, 9.033715685419241674536975332396
