| L(s) = 1 | − 3.94·2-s + 7.56·4-s − 16.6·7-s + 1.72·8-s + 11·11-s + 79.9·13-s + 65.8·14-s − 67.3·16-s + 5.54·17-s + 24.4·19-s − 43.3·22-s − 120.·23-s − 315.·26-s − 126.·28-s + 125.·29-s − 139.·31-s + 251.·32-s − 21.8·34-s + 394.·37-s − 96.3·38-s + 314.·41-s + 261.·43-s + 83.1·44-s + 477.·46-s + 156.·47-s − 64.6·49-s + 604.·52-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.945·4-s − 0.900·7-s + 0.0763·8-s + 0.301·11-s + 1.70·13-s + 1.25·14-s − 1.05·16-s + 0.0790·17-s + 0.294·19-s − 0.420·22-s − 1.09·23-s − 2.37·26-s − 0.851·28-s + 0.803·29-s − 0.808·31-s + 1.39·32-s − 0.110·34-s + 1.75·37-s − 0.411·38-s + 1.19·41-s + 0.927·43-s + 0.285·44-s + 1.52·46-s + 0.484·47-s − 0.188·49-s + 1.61·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9668314181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9668314181\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 3.94T + 8T^{2} \) |
| 7 | \( 1 + 16.6T + 343T^{2} \) |
| 13 | \( 1 - 79.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 394.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 314.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 261.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 489.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 247.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 774.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 687.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 942.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 655.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 20.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.32e3T + 9.12e5T^{2} \) |
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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
−8.899356114808172902017771539376, −7.87792057464148195912964070302, −7.36903190787565635806124105539, −6.20493146873989827286378012430, −6.03045185718292937640020754860, −4.42684875419661707892024840446, −3.65103091779852142681462206678, −2.54422571673684276195305132826, −1.36558471601778341714279961866, −0.59634520962853773022544656796,
0.59634520962853773022544656796, 1.36558471601778341714279961866, 2.54422571673684276195305132826, 3.65103091779852142681462206678, 4.42684875419661707892024840446, 6.03045185718292937640020754860, 6.20493146873989827286378012430, 7.36903190787565635806124105539, 7.87792057464148195912964070302, 8.899356114808172902017771539376
