| L(s) = 1 | − 4.74·2-s + 14.4·4-s − 4.51·5-s − 7.48·7-s − 30.7·8-s + 21.4·10-s + 66.8·11-s − 13·13-s + 35.4·14-s + 29.8·16-s − 96.9·17-s + 31.4·19-s − 65.4·20-s − 317.·22-s − 183.·23-s − 104.·25-s + 61.6·26-s − 108.·28-s − 112.·29-s − 77.2·31-s + 104.·32-s + 459.·34-s + 33.7·35-s + 54.7·37-s − 149.·38-s + 138.·40-s − 451.·41-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.81·4-s − 0.403·5-s − 0.404·7-s − 1.35·8-s + 0.677·10-s + 1.83·11-s − 0.277·13-s + 0.677·14-s + 0.467·16-s − 1.38·17-s + 0.380·19-s − 0.731·20-s − 3.07·22-s − 1.66·23-s − 0.836·25-s + 0.464·26-s − 0.731·28-s − 0.718·29-s − 0.447·31-s + 0.575·32-s + 2.31·34-s + 0.163·35-s + 0.243·37-s − 0.637·38-s + 0.548·40-s − 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 4.74T + 8T^{2} \) |
| 5 | \( 1 + 4.51T + 125T^{2} \) |
| 7 | \( 1 + 7.48T + 343T^{2} \) |
| 11 | \( 1 - 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 54.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 822.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 545.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 230.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93035959339855951985913242629, −11.42552329360681036081902905990, −10.08319105319050334074844463645, −9.294077543147806545208936945520, −8.433134935920717875958997297661, −7.21056578691358615986836758512, −6.32570344893574209253582398687, −3.95082809222005994297292905025, −1.82422593312439725647634377140, 0,
1.82422593312439725647634377140, 3.95082809222005994297292905025, 6.32570344893574209253582398687, 7.21056578691358615986836758512, 8.433134935920717875958997297661, 9.294077543147806545208936945520, 10.08319105319050334074844463645, 11.42552329360681036081902905990, 11.93035959339855951985913242629
