| L(s) = 1 | − 4.95·2-s + 16.5·4-s + 16.1·5-s − 7·7-s − 42.1·8-s − 79.9·10-s − 30.6·11-s − 78.0·13-s + 34.6·14-s + 76.7·16-s + 106.·17-s − 49.5·19-s + 266.·20-s + 151.·22-s + 39.0·23-s + 135.·25-s + 386.·26-s − 115.·28-s + 5.74·29-s + 184.·31-s − 42.5·32-s − 529.·34-s − 112.·35-s + 91.0·37-s + 245.·38-s − 680.·40-s − 82.8·41-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 2.06·4-s + 1.44·5-s − 0.377·7-s − 1.86·8-s − 2.52·10-s − 0.839·11-s − 1.66·13-s + 0.661·14-s + 1.19·16-s + 1.52·17-s − 0.597·19-s + 2.98·20-s + 1.47·22-s + 0.353·23-s + 1.08·25-s + 2.91·26-s − 0.780·28-s + 0.0368·29-s + 1.06·31-s − 0.235·32-s − 2.67·34-s − 0.545·35-s + 0.404·37-s + 1.04·38-s − 2.69·40-s − 0.315·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8980394472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8980394472\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 4.95T + 8T^{2} \) |
| 5 | \( 1 - 16.1T + 125T^{2} \) |
| 11 | \( 1 + 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 39.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.74T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 82.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 91.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 237.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 4.20T + 2.05e5T^{2} \) |
| 61 | \( 1 - 626.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 678.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 747.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 23.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 154.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 282.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 111.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−9.977310175743996641667954246812, −9.763983323610604882221938588905, −8.761636538101372553722713065210, −7.78955210060696461456792752835, −7.02790630914707142203754381631, −6.03158612220103411059257095882, −5.09005012263719131677255340872, −2.79481786995232292735474777423, −2.09366544717222594117179156442, −0.72642467422810855322448811170,
0.72642467422810855322448811170, 2.09366544717222594117179156442, 2.79481786995232292735474777423, 5.09005012263719131677255340872, 6.03158612220103411059257095882, 7.02790630914707142203754381631, 7.78955210060696461456792752835, 8.761636538101372553722713065210, 9.763983323610604882221938588905, 9.977310175743996641667954246812
