| L(s) = 1 | + (15.7 − 3.06i)2-s + (−13.4 − 32.5i)3-s + (237. − 96.1i)4-s + (−108. + 262. i)5-s + (−311. − 470. i)6-s + (1.24e3 − 1.24e3i)7-s + (3.43e3 − 2.23e3i)8-s + (3.75e3 − 3.75e3i)9-s + (−902. + 4.44e3i)10-s + (347. − 839. i)11-s + (−6.33e3 − 6.43e3i)12-s + (−1.19e4 − 2.88e4i)13-s + (1.57e4 − 2.33e4i)14-s + 1.00e4·15-s + (4.70e4 − 4.56e4i)16-s − 3.22e4i·17-s + ⋯ |
| L(s) = 1 | + (0.981 − 0.191i)2-s + (−0.166 − 0.402i)3-s + (0.926 − 0.375i)4-s + (−0.173 + 0.419i)5-s + (−0.240 − 0.363i)6-s + (0.519 − 0.519i)7-s + (0.837 − 0.545i)8-s + (0.572 − 0.572i)9-s + (−0.0902 + 0.444i)10-s + (0.0237 − 0.0573i)11-s + (−0.305 − 0.310i)12-s + (−0.418 − 1.00i)13-s + (0.410 − 0.608i)14-s + 0.197·15-s + (0.717 − 0.696i)16-s − 0.385i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.69945 - 1.84158i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.69945 - 1.84158i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.7 + 3.06i)T \) |
| good | 3 | \( 1 + (13.4 + 32.5i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (108. - 262. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-1.24e3 + 1.24e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (-347. + 839. i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (1.19e4 + 2.88e4i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 + 3.22e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (3.67e4 - 1.52e4i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (-1.33e5 - 1.33e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (-5.67e5 + 2.34e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 4.50e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (8.82e5 - 2.13e6i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (9.91e5 - 9.91e5i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (2.24e6 - 5.41e6i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 1.22e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-5.86e6 - 2.42e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (1.20e7 + 4.97e6i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-5.36e6 + 2.22e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-7.80e5 - 1.88e6i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-9.12e6 + 9.12e6i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (-3.28e7 + 3.28e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 5.48e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (3.60e7 - 1.49e7i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-5.53e7 - 5.53e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 - 3.01e7T + 7.83e15T^{2} \) |
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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
−14.71744118949203373862705876250, −13.46766716403015444633757792137, −12.42554069316762111766800849088, −11.27076532550974758051529954347, −10.09637039249913527777108637137, −7.65156438175217012004823953935, −6.55192961900206445110528220358, −4.84232625731143623763059994920, −3.18517662326646328596541247761, −1.18017281797353602551631749601,
2.05147417680349595747715878884, 4.23749823787932435256809355588, 5.21360172344258854601249161123, 6.97673839385341128751092483706, 8.569513969411125807323010229692, 10.49384600319985779796131286965, 11.75713040352612895426996695725, 12.77121264211825414285550298639, 14.11996754080600719696603766669, 15.20341830863650104644675115585
