| L(s) = 1 | + (16 − 27.7i)2-s + (198 + 342. i)3-s + (−511. − 886. i)4-s + (−3.67e3 + 6.36e3i)5-s + 1.26e4·6-s − 3.27e4·8-s + (1.01e4 − 1.76e4i)9-s + (1.17e5 + 2.03e5i)10-s + (5.43e4 + 9.42e4i)11-s + (2.02e5 − 3.51e5i)12-s − 6.35e5·13-s − 2.91e6·15-s + (−5.24e5 + 9.08e5i)16-s + (4.61e6 + 7.98e6i)17-s + (−3.25e5 − 5.63e5i)18-s + (3.77e6 − 6.54e6i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.470 + 0.814i)3-s + (−0.249 − 0.433i)4-s + (−0.525 + 0.910i)5-s + 0.665·6-s − 0.353·8-s + (0.0573 − 0.0993i)9-s + (0.371 + 0.644i)10-s + (0.101 + 0.176i)11-s + (0.235 − 0.407i)12-s − 0.474·13-s − 0.989·15-s + (−0.125 + 0.216i)16-s + (0.787 + 1.36i)17-s + (−0.0405 − 0.0702i)18-s + (0.350 − 0.606i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0436792 + 0.688294i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0436792 + 0.688294i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 + 27.7i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-198 - 342. i)T + (-8.85e4 + 1.53e5i)T^{2} \) |
| 5 | \( 1 + (3.67e3 - 6.36e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-5.43e4 - 9.42e4i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 6.35e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-4.61e6 - 7.98e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-3.77e6 + 6.54e6i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (1.32e7 - 2.29e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.69e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (-2.56e7 - 4.44e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.25e8 + 2.17e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 + 9.28e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.81e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (2.61e8 - 4.53e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (2.09e9 + 3.63e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (4.57e9 + 7.91e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-3.31e9 + 5.74e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-1.43e9 - 2.49e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 4.34e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (1.17e10 + 2.03e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.43e10 + 2.49e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + 5.57e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (3.90e10 - 6.75e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 + 2.66e10T + 7.15e21T^{2} \) |
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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
−12.07355809141780618224645707625, −11.09013992317102190255574878468, −10.14949559104615660437706019124, −9.381826504303463811702612258494, −7.952930583245054261639497992720, −6.63688777981012910520726332964, −5.08981463465020825005452020371, −3.70811495409529927056634936949, −3.28844534577763227841631674490, −1.73098224192820820123744935683,
0.13029375810988966980444141046, 1.41649143807625743093423510703, 2.95077030327306412697961239505, 4.42351257992594863644330405844, 5.45822668712970180648169818251, 6.99067654061842791934189913511, 7.81844737367620168425036187160, 8.584458284791005107429563571059, 9.864057538543247686998164430025, 11.74041171752136151184049708541
