| L(s) = 1 | − 2.28·2-s − 2.78·4-s + 5·5-s + 13.7·7-s + 24.6·8-s − 11.4·10-s + 11·11-s + 20.3·13-s − 31.2·14-s − 33.9·16-s + 25.5·17-s + 77.9·19-s − 13.9·20-s − 25.1·22-s − 203.·23-s + 25·25-s − 46.4·26-s − 38.2·28-s + 46.1·29-s + 167.·31-s − 119.·32-s − 58.2·34-s + 68.5·35-s − 244.·37-s − 177.·38-s + 123.·40-s − 204.·41-s + ⋯ |
| L(s) = 1 | − 0.807·2-s − 0.348·4-s + 0.447·5-s + 0.740·7-s + 1.08·8-s − 0.360·10-s + 0.301·11-s + 0.434·13-s − 0.597·14-s − 0.530·16-s + 0.364·17-s + 0.941·19-s − 0.155·20-s − 0.243·22-s − 1.84·23-s + 0.200·25-s − 0.350·26-s − 0.257·28-s + 0.295·29-s + 0.969·31-s − 0.660·32-s − 0.293·34-s + 0.331·35-s − 1.08·37-s − 0.759·38-s + 0.486·40-s − 0.779·41-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
| \(L(2)\) |
\(\approx\) |
\(1.379502045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.379502045\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.28T + 8T^{2} \) |
| 7 | \( 1 - 13.7T + 343T^{2} \) |
| 13 | \( 1 - 20.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 203.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 46.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 204.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 489.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 324.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 381.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 819.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 867.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 719.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 345.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 583.T + 9.12e5T^{2} \) |
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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
−10.16369287089547131021106089877, −9.787228159063945682284031911992, −8.635878868365091459860725414148, −8.132134538005154642517762155227, −7.12174311232539885700579531238, −5.84290312686592146414585720855, −4.86082744529198631643308094008, −3.74916713444470352812458445060, −1.96776890846808509658824204628, −0.878310799682580610545389581467,
0.878310799682580610545389581467, 1.96776890846808509658824204628, 3.74916713444470352812458445060, 4.86082744529198631643308094008, 5.84290312686592146414585720855, 7.12174311232539885700579531238, 8.132134538005154642517762155227, 8.635878868365091459860725414148, 9.787228159063945682284031911992, 10.16369287089547131021106089877
