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
−14.01479746143673509093877879861, −12.83888521721007268206101016327, −11.92486911878858546636421911050, −11.37552368827229512107900388160, −9.294185262918320544945765156512, −7.937558727345570531158057596464, −6.63500147609821805236398925858, −4.63226694866319387786640878159, −1.74429029708867326847395902448, −0.29072283200364142876419307111,
3.84228528712638409094515248200, 5.37124460745713693042800703428, 6.85767750955604901901666167323, 8.588001348959135604777045145704, 10.24315454062486884497958423251, 10.73405357561680588196854905388, 12.16541181784130442546102664947, 14.50542176833460265583136732751, 15.05139362650192189790965160657, 16.07806960689714893448032594316