| L(s) = 1 | + (−2 + 3.46i)2-s + (−12.1 + 21.0i)3-s + (−7.99 − 13.8i)4-s + (−28.6 + 49.6i)5-s + (−48.6 − 84.3i)6-s + 33.7·7-s + 63.9·8-s + (−174. − 302. i)9-s + (−114. − 198. i)10-s + 296.·11-s + 389.·12-s + (−360. − 624. i)13-s + (−67.4 + 116. i)14-s + (−697. − 1.20e3i)15-s + (−128 + 221. i)16-s + (−333. + 577. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.780 + 1.35i)3-s + (−0.249 − 0.433i)4-s + (−0.512 + 0.887i)5-s + (−0.552 − 0.956i)6-s + 0.260·7-s + 0.353·8-s + (−0.718 − 1.24i)9-s + (−0.362 − 0.627i)10-s + 0.739·11-s + 0.780·12-s + (−0.591 − 1.02i)13-s + (−0.0920 + 0.159i)14-s + (−0.800 − 1.38i)15-s + (−0.125 + 0.216i)16-s + (−0.279 + 0.484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.227441 - 0.338805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227441 - 0.338805i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 3.46i)T \) |
| 19 | \( 1 + (890. - 1.29e3i)T \) |
| good | 3 | \( 1 + (12.1 - 21.0i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (28.6 - 49.6i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 33.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 296.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (360. + 624. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (333. - 577. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (1.33e3 + 2.30e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-3.76e3 - 6.51e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 2.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.51e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (924. - 1.60e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.06e4 + 1.83e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.16e4 + 2.01e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-6.68e3 - 1.15e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (3.99e3 - 6.92e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.41e4 - 4.19e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.00e4 - 1.73e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (8.01e3 - 1.38e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.72e4 - 2.98e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.14e4 - 5.44e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 8.12e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.59e4 - 4.49e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (8.57e4 - 1.48e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−16.07806960689714893448032594316, −15.05139362650192189790965160657, −14.50542176833460265583136732751, −12.16541181784130442546102664947, −10.73405357561680588196854905388, −10.24315454062486884497958423251, −8.588001348959135604777045145704, −6.85767750955604901901666167323, −5.37124460745713693042800703428, −3.84228528712638409094515248200,
0.29072283200364142876419307111, 1.74429029708867326847395902448, 4.63226694866319387786640878159, 6.63500147609821805236398925858, 7.937558727345570531158057596464, 9.294185262918320544945765156512, 11.37552368827229512107900388160, 11.92486911878858546636421911050, 12.83888521721007268206101016327, 14.01479746143673509093877879861
