| L(s) = 1 | − 4.41·2-s − 3.24·3-s + 11.4·4-s − 5·5-s + 14.3·6-s − 15.3·8-s − 16.4·9-s + 22.0·10-s − 0.142·11-s − 37.2·12-s + 32.1·13-s + 16.2·15-s − 23.9·16-s + 114.·17-s + 72.7·18-s − 43.2·19-s − 57.4·20-s + 0.627·22-s + 154.·23-s + 49.8·24-s + 25·25-s − 141.·26-s + 141.·27-s − 40.1·29-s − 71.5·30-s − 75.4·31-s + 228.·32-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.624·3-s + 1.43·4-s − 0.447·5-s + 0.973·6-s − 0.679·8-s − 0.610·9-s + 0.697·10-s − 0.00389·11-s − 0.895·12-s + 0.685·13-s + 0.279·15-s − 0.374·16-s + 1.63·17-s + 0.952·18-s − 0.521·19-s − 0.642·20-s + 0.00608·22-s + 1.40·23-s + 0.424·24-s + 0.200·25-s − 1.07·26-s + 1.00·27-s − 0.257·29-s − 0.435·30-s − 0.436·31-s + 1.26·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.41T + 8T^{2} \) |
| 3 | \( 1 + 3.24T + 27T^{2} \) |
| 11 | \( 1 + 0.142T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 40.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 75.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 400.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 95.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 7.49T + 1.03e5T^{2} \) |
| 53 | \( 1 + 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 796.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 757.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 37.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 80.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 317.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 945.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 783.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 393.T + 9.12e5T^{2} \) |
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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
−10.97270330664543862439346523607, −10.28061049917572038710589505732, −9.105818861746658644175725822508, −8.367496780956048721166456507188, −7.43586182918323600288200595072, −6.37438052289435044536236668632, −5.12052218472154734965649134747, −3.24898798493019027963937352291, −1.31414350096008295868992079198, 0,
1.31414350096008295868992079198, 3.24898798493019027963937352291, 5.12052218472154734965649134747, 6.37438052289435044536236668632, 7.43586182918323600288200595072, 8.367496780956048721166456507188, 9.105818861746658644175725822508, 10.28061049917572038710589505732, 10.97270330664543862439346523607
