| L(s) = 1 | − 3.54·2-s + 9·3-s − 19.4·4-s − 92.1·5-s − 31.8·6-s − 8.98·7-s + 182.·8-s + 81·9-s + 326.·10-s + 612.·11-s − 175.·12-s − 496.·13-s + 31.8·14-s − 829.·15-s − 23.4·16-s + 745.·17-s − 286.·18-s + 1.29e3·19-s + 1.79e3·20-s − 80.9·21-s − 2.17e3·22-s − 529·23-s + 1.64e3·24-s + 5.36e3·25-s + 1.75e3·26-s + 729·27-s + 174.·28-s + ⋯ |
| L(s) = 1 | − 0.626·2-s + 0.577·3-s − 0.607·4-s − 1.64·5-s − 0.361·6-s − 0.0693·7-s + 1.00·8-s + 0.333·9-s + 1.03·10-s + 1.52·11-s − 0.350·12-s − 0.814·13-s + 0.0434·14-s − 0.951·15-s − 0.0228·16-s + 0.625·17-s − 0.208·18-s + 0.825·19-s + 1.00·20-s − 0.0400·21-s − 0.956·22-s − 0.208·23-s + 0.581·24-s + 1.71·25-s + 0.510·26-s + 0.192·27-s + 0.0421·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9664536085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9664536085\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 3.54T + 32T^{2} \) |
| 5 | \( 1 + 92.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 8.98T + 1.68e4T^{2} \) |
| 11 | \( 1 - 612.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 496.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 745.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.29e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 840.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.53T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.99e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.37e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.71e4T + 8.58e9T^{2} \) |
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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
−14.02444355149951149352689985262, −12.42782918073078516525213616656, −11.63122173781117147315355718384, −10.05818148688829285093997454709, −8.985329741924781868167586931587, −8.037280753225211284468601151860, −7.12167511553267171037756707280, −4.57896778918635400770656627339, −3.53511996991014255088625359725, −0.860939740144385835086401300255,
0.860939740144385835086401300255, 3.53511996991014255088625359725, 4.57896778918635400770656627339, 7.12167511553267171037756707280, 8.037280753225211284468601151860, 8.985329741924781868167586931587, 10.05818148688829285093997454709, 11.63122173781117147315355718384, 12.42782918073078516525213616656, 14.02444355149951149352689985262
