| L(s) = 1 | + (−16.3 + 28.2i)2-s + (−65.4 + 113. i)3-s + (−276. − 479. i)4-s + (−805. + 1.39e3i)5-s + (−2.13e3 − 3.69e3i)6-s − 7.03e3·7-s + 1.35e3·8-s + (1.28e3 + 2.22e3i)9-s + (−2.63e4 − 4.55e4i)10-s + 4.81e4·11-s + 7.24e4·12-s + (7.81e4 + 1.35e5i)13-s + (1.14e5 − 1.98e5i)14-s + (−1.05e5 − 1.82e5i)15-s + (1.19e5 − 2.07e5i)16-s + (1.15e5 − 1.99e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.721 + 1.24i)2-s + (−0.466 + 0.807i)3-s + (−0.540 − 0.936i)4-s + (−0.576 + 0.998i)5-s + (−0.672 − 1.16i)6-s − 1.10·7-s + 0.117·8-s + (0.0652 + 0.112i)9-s + (−0.831 − 1.44i)10-s + 0.991·11-s + 1.00·12-s + (0.758 + 1.31i)13-s + (0.798 − 1.38i)14-s + (−0.537 − 0.931i)15-s + (0.456 − 0.789i)16-s + (0.334 − 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.829i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.559 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.346290 - 0.184117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.346290 - 0.184117i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (3.51e5 - 4.46e5i)T \) |
| good | 2 | \( 1 + (16.3 - 28.2i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (65.4 - 113. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (805. - 1.39e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + 7.03e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.81e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-7.81e4 - 1.35e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.15e5 + 1.99e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (2.52e5 + 4.38e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (5.23e5 + 9.07e5i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 - 1.56e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.01e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.04e7 + 1.81e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.96e5 + 3.39e5i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (2.38e7 + 4.13e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-4.25e7 - 7.36e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (6.47e7 - 1.12e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-5.02e6 - 8.70e6i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.06e8 - 1.84e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.78e8 + 3.09e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (1.52e8 - 2.64e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.42e8 + 4.20e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 4.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (3.82e8 + 6.61e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (5.47e8 - 9.48e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−16.87605578520489578863091545502, −16.27287533053999309730908432475, −15.37247252431691482875385616373, −14.15837858224057062174150117878, −11.75753120706558390755565054208, −10.25400093784305963279135216660, −9.027114158851272188805944608138, −7.13379451806928037335199663603, −6.17743836973926527224800350802, −3.80799712554460070753098260387,
0.29236378716312148807936946767, 1.21523999006314592894548095127, 3.54597487811084123543715039625, 6.29760818375700884238803275568, 8.391662539132724471484612133718, 9.672094242323586923615070785131, 11.27755342578062420313752338780, 12.54878433074059546676539446145, 12.86063986419711518297595852573, 15.55650921783302064902948480259
