| L(s) = 1 | + 1.29·2-s − 6.32·4-s − 5·5-s − 23.9·7-s − 18.5·8-s − 6.46·10-s − 11·11-s + 11.9·13-s − 30.9·14-s + 26.6·16-s + 94.7·17-s − 139.·19-s + 31.6·20-s − 14.2·22-s − 85.7·23-s + 25·25-s + 15.4·26-s + 151.·28-s + 194.·29-s + 220.·31-s + 182.·32-s + 122.·34-s + 119.·35-s + 323.·37-s − 179.·38-s + 92.6·40-s − 146.·41-s + ⋯ |
| L(s) = 1 | + 0.456·2-s − 0.791·4-s − 0.447·5-s − 1.29·7-s − 0.818·8-s − 0.204·10-s − 0.301·11-s + 0.254·13-s − 0.590·14-s + 0.417·16-s + 1.35·17-s − 1.68·19-s + 0.353·20-s − 0.137·22-s − 0.777·23-s + 0.200·25-s + 0.116·26-s + 1.02·28-s + 1.24·29-s + 1.27·31-s + 1.00·32-s + 0.617·34-s + 0.578·35-s + 1.43·37-s − 0.767·38-s + 0.366·40-s − 0.557·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.077918614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.077918614\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 1.29T + 8T^{2} \) |
| 7 | \( 1 + 23.9T + 343T^{2} \) |
| 13 | \( 1 - 11.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 85.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 323.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 48.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 298.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 504.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 267.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 568.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 647.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 895.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 894.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−10.19631014066603988222623952398, −9.881278874516665038012639459806, −8.640707932153853586991835577744, −8.026344051237433981142205442080, −6.57786813795161654997581982079, −5.91033054166529855533595808358, −4.64922841933483560066788901331, −3.75036016503333606783408325684, −2.82900517970343742674727685840, −0.58878018749735733247057542432,
0.58878018749735733247057542432, 2.82900517970343742674727685840, 3.75036016503333606783408325684, 4.64922841933483560066788901331, 5.91033054166529855533595808358, 6.57786813795161654997581982079, 8.026344051237433981142205442080, 8.640707932153853586991835577744, 9.881278874516665038012639459806, 10.19631014066603988222623952398
