| L(s) = 1 | + (−24.5 − 2.76i)2-s + (−135. − 84.9i)3-s + (345. + 78.8i)4-s + (−399. + 318. i)5-s + (3.08e3 + 2.45e3i)6-s + (40.6 + 177. i)7-s + (−2.29e3 − 802. i)8-s + (8.21e3 + 1.70e4i)9-s + (1.06e4 − 6.70e3i)10-s + (−2.02e4 + 7.07e3i)11-s + (−3.99e4 − 3.99e4i)12-s + (1.91e3 − 3.98e3i)13-s + (−504. − 4.47e3i)14-s + (8.09e4 − 9.12e3i)15-s + (−2.76e4 − 1.33e4i)16-s + (2.16e4 − 2.16e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.53 − 0.172i)2-s + (−1.66 − 1.04i)3-s + (1.34 + 0.307i)4-s + (−0.638 + 0.509i)5-s + (2.37 + 1.89i)6-s + (0.0169 + 0.0740i)7-s + (−0.559 − 0.195i)8-s + (1.25 + 2.59i)9-s + (1.06 − 0.670i)10-s + (−1.38 + 0.483i)11-s + (−1.92 − 1.92i)12-s + (0.0671 − 0.139i)13-s + (−0.0131 − 0.116i)14-s + (1.59 − 0.180i)15-s + (−0.422 − 0.203i)16-s + (0.259 − 0.259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0719439 - 0.0859474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0719439 - 0.0859474i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (5.87e5 - 3.93e5i)T \) |
| good | 2 | \( 1 + (24.5 + 2.76i)T + (249. + 56.9i)T^{2} \) |
| 3 | \( 1 + (135. + 84.9i)T + (2.84e3 + 5.91e3i)T^{2} \) |
| 5 | \( 1 + (399. - 318. i)T + (8.69e4 - 3.80e5i)T^{2} \) |
| 7 | \( 1 + (-40.6 - 177. i)T + (-5.19e6 + 2.50e6i)T^{2} \) |
| 11 | \( 1 + (2.02e4 - 7.07e3i)T + (1.67e8 - 1.33e8i)T^{2} \) |
| 13 | \( 1 + (-1.91e3 + 3.98e3i)T + (-5.08e8 - 6.37e8i)T^{2} \) |
| 17 | \( 1 + (-2.16e4 + 2.16e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (-3.21e4 - 5.11e4i)T + (-7.36e9 + 1.53e10i)T^{2} \) |
| 23 | \( 1 + (2.75e5 - 3.45e5i)T + (-1.74e10 - 7.63e10i)T^{2} \) |
| 31 | \( 1 + (1.06e6 + 1.19e5i)T + (8.31e11 + 1.89e11i)T^{2} \) |
| 37 | \( 1 + (-1.72e6 - 6.04e5i)T + (2.74e12 + 2.18e12i)T^{2} \) |
| 41 | \( 1 + (-1.18e6 - 1.18e6i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (5.93e5 + 5.27e6i)T + (-1.13e13 + 2.60e12i)T^{2} \) |
| 47 | \( 1 + (1.04e6 + 2.99e6i)T + (-1.86e13 + 1.48e13i)T^{2} \) |
| 53 | \( 1 + (3.26e6 + 4.09e6i)T + (-1.38e13 + 6.06e13i)T^{2} \) |
| 59 | \( 1 + 1.03e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + (5.71e6 + 3.59e6i)T + (8.31e13 + 1.72e14i)T^{2} \) |
| 67 | \( 1 + (-5.85e6 - 1.21e7i)T + (-2.53e14 + 3.17e14i)T^{2} \) |
| 71 | \( 1 + (1.91e7 - 3.97e7i)T + (-4.02e14 - 5.04e14i)T^{2} \) |
| 73 | \( 1 + (3.23e6 - 3.64e5i)T + (7.86e14 - 1.79e14i)T^{2} \) |
| 79 | \( 1 + (-1.29e7 + 3.71e7i)T + (-1.18e15 - 9.45e14i)T^{2} \) |
| 83 | \( 1 + (-1.24e7 + 5.47e7i)T + (-2.02e15 - 9.77e14i)T^{2} \) |
| 89 | \( 1 + (-2.84e7 - 3.20e6i)T + (3.83e15 + 8.75e14i)T^{2} \) |
| 97 | \( 1 + (4.20e7 - 2.64e7i)T + (3.40e15 - 7.06e15i)T^{2} \) |
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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
−15.71109134379576989192489662657, −13.20323894539514639391048917450, −11.87356273451085921413211331584, −11.04231336530212421052032338491, −10.11382643304741597084658481605, −7.73285574088030476575555139504, −7.29931066382314197933037102916, −5.49195674310138053615133052419, −1.82069871702762898895936569376, −0.20132523703559979757544266770,
0.59298473406131408196489725688, 4.44901571347751932247186611838, 6.02263991005889726915397959774, 7.80362815325965379082015267587, 9.348858604828791259216910177823, 10.51138407586983755429226389785, 11.18596759339130743218185557866, 12.53045635260814200619279316717, 15.39366292676712134348849324601, 16.28006089476209142700478499866
