| L(s) = 1 | + (−121.5 − 210. i)3-s + (3.23e3 − 5.61e3i)5-s + (−4.40e4 + 6.34e3i)7-s + (−2.95e4 + 5.11e4i)9-s + (1.53e5 + 2.66e5i)11-s + 5.84e5·13-s − 1.57e6·15-s + (4.72e6 + 8.17e6i)17-s + (8.44e6 − 1.46e7i)19-s + (6.68e6 + 8.49e6i)21-s + (1.59e7 − 2.76e7i)23-s + (3.42e6 + 5.93e6i)25-s + 1.43e7·27-s − 1.08e8·29-s + (1.25e8 + 2.17e8i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.463 − 0.802i)5-s + (−0.989 + 0.142i)7-s + (−0.166 + 0.288i)9-s + (0.288 + 0.498i)11-s + 0.436·13-s − 0.535·15-s + (0.806 + 1.39i)17-s + (0.782 − 1.35i)19-s + (0.357 + 0.453i)21-s + (0.516 − 0.894i)23-s + (0.0701 + 0.121i)25-s + 0.192·27-s − 0.977·29-s + (0.787 + 1.36i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.41274 - 1.14730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41274 - 1.14730i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (121.5 + 210. i)T \) |
| 7 | \( 1 + (4.40e4 - 6.34e3i)T \) |
| good | 5 | \( 1 + (-3.23e3 + 5.61e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-1.53e5 - 2.66e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 5.84e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-4.72e6 - 8.17e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.44e6 + 1.46e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-1.59e7 + 2.76e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.08e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.25e8 - 2.17e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-8.03e7 + 1.39e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 5.43e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.35e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-4.31e8 + 7.46e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.83e9 + 3.18e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (1.91e9 + 3.31e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.30e9 - 3.98e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-9.21e8 - 1.59e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.51e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + (1.51e10 + 2.61e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.74e10 + 3.02e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 2.03e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-4.42e10 + 7.65e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 9.58e10T + 7.15e21T^{2} \) |
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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
−12.10826991197563842164849771221, −10.66876399240481091375947671183, −9.489372397842800209408215674773, −8.579550083964866414203406568967, −7.06002585718983060317039991565, −6.05834038930871163370614074798, −4.92557212993620973130870171171, −3.27679666622795953066021362686, −1.69253748101229580576123674567, −0.60250830392242565632176025918,
0.937409976083912145973097374341, 2.86457206099333813264607946752, 3.71221684598208159910139681037, 5.52486434935055538176042803467, 6.36754870568694421433273421468, 7.61335321289871134454914209234, 9.400845075560228057189948973939, 9.945346332370011279299472519182, 11.09278458326684137201910840847, 12.08444859155380557166069128431
