| L(s) = 1 | + 3.04·2-s + 1.25·4-s − 5·5-s − 13.7·7-s − 20.5·8-s − 15.2·10-s + 31.8·11-s + 58.2·13-s − 41.7·14-s − 72.4·16-s + 109.·17-s + 129.·19-s − 6.26·20-s + 96.7·22-s + 79.6·23-s + 25·25-s + 177.·26-s − 17.1·28-s + 9.03·29-s − 33.3·31-s − 56.1·32-s + 331.·34-s + 68.5·35-s − 22.1·37-s + 394.·38-s + 102.·40-s + 121.·41-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.156·4-s − 0.447·5-s − 0.740·7-s − 0.907·8-s − 0.480·10-s + 0.871·11-s + 1.24·13-s − 0.796·14-s − 1.13·16-s + 1.55·17-s + 1.56·19-s − 0.0699·20-s + 0.937·22-s + 0.722·23-s + 0.200·25-s + 1.33·26-s − 0.115·28-s + 0.0578·29-s − 0.193·31-s − 0.310·32-s + 1.67·34-s + 0.331·35-s − 0.0984·37-s + 1.68·38-s + 0.405·40-s + 0.463·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.816280121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.816280121\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 2 | \( 1 - 3.04T + 8T^{2} \) |
| 7 | \( 1 + 13.7T + 343T^{2} \) |
| 11 | \( 1 - 31.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 79.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 9.03T + 2.43e4T^{2} \) |
| 31 | \( 1 + 33.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 121.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 10.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 441.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 593.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 442.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 144.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 862.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 818.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 495.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 424.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.59e3T + 9.12e5T^{2} \) |
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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
−11.22165440255486708848627151630, −9.804374744384881384319713696459, −9.134622147112753285814281558174, −7.993417134269002912516042296422, −6.76236213442568382226804571235, −5.91574438080211578171159862908, −4.95968296431347293559009230132, −3.55269551529212876575052556696, −3.33075536230368044243124874591, −0.987617973996538241153943980658,
0.987617973996538241153943980658, 3.33075536230368044243124874591, 3.55269551529212876575052556696, 4.95968296431347293559009230132, 5.91574438080211578171159862908, 6.76236213442568382226804571235, 7.993417134269002912516042296422, 9.134622147112753285814281558174, 9.804374744384881384319713696459, 11.22165440255486708848627151630
