| L(s) = 1 | + (−7.27 − 8.66i)2-s + (7.33 + 80.6i)3-s + (−22.2 + 126. i)4-s + (58.6 + 161. i)5-s + (645. − 650. i)6-s + (−371. − 2.10e3i)7-s + (1.25e3 − 724. i)8-s + (−6.45e3 + 1.18e3i)9-s + (969. − 1.67e3i)10-s + (6.80e3 − 1.87e4i)11-s + (−1.03e4 − 867. i)12-s + (−1.07e3 − 898. i)13-s + (−1.55e4 + 1.85e4i)14-s + (−1.25e4 + 5.91e3i)15-s + (−1.53e4 − 5.60e3i)16-s + (5.04e4 + 2.91e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (0.0905 + 0.995i)3-s + (−0.0868 + 0.492i)4-s + (0.0938 + 0.257i)5-s + (0.498 − 0.501i)6-s + (−0.154 − 0.877i)7-s + (0.306 − 0.176i)8-s + (−0.983 + 0.180i)9-s + (0.0969 − 0.167i)10-s + (0.465 − 1.27i)11-s + (−0.498 − 0.0418i)12-s + (−0.0374 − 0.0314i)13-s + (−0.405 + 0.482i)14-s + (−0.248 + 0.116i)15-s + (−0.234 − 0.0855i)16-s + (0.604 + 0.348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.49499 - 0.144914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49499 - 0.144914i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.27 + 8.66i)T \) |
| 3 | \( 1 + (-7.33 - 80.6i)T \) |
| good | 5 | \( 1 + (-58.6 - 161. i)T + (-2.99e5 + 2.51e5i)T^{2} \) |
| 7 | \( 1 + (371. + 2.10e3i)T + (-5.41e6 + 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-6.80e3 + 1.87e4i)T + (-1.64e8 - 1.37e8i)T^{2} \) |
| 13 | \( 1 + (1.07e3 + 898. i)T + (1.41e8 + 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-5.04e4 - 2.91e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-8.93e4 - 1.54e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.28e5 + 2.26e4i)T + (7.35e10 + 2.67e10i)T^{2} \) |
| 29 | \( 1 + (-7.69e5 - 9.17e5i)T + (-8.68e10 + 4.92e11i)T^{2} \) |
| 31 | \( 1 + (-1.38e5 + 7.84e5i)T + (-8.01e11 - 2.91e11i)T^{2} \) |
| 37 | \( 1 + (-1.31e6 + 2.27e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (2.32e5 - 2.77e5i)T + (-1.38e12 - 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-4.97e6 - 1.81e6i)T + (8.95e12 + 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-1.90e5 + 3.35e4i)T + (2.23e13 - 8.14e12i)T^{2} \) |
| 53 | \( 1 - 8.84e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-3.69e5 - 1.01e6i)T + (-1.12e14 + 9.43e13i)T^{2} \) |
| 61 | \( 1 + (-1.22e6 - 6.96e6i)T + (-1.80e14 + 6.55e13i)T^{2} \) |
| 67 | \( 1 + (-5.96e5 - 5.00e5i)T + (7.05e13 + 3.99e14i)T^{2} \) |
| 71 | \( 1 + (2.19e7 + 1.26e7i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.40e7 + 2.43e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-5.12e7 + 4.29e7i)T + (2.63e14 - 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-3.12e7 - 3.71e7i)T + (-3.91e14 + 2.21e15i)T^{2} \) |
| 89 | \( 1 + (-2.76e7 + 1.59e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (1.10e7 + 4.02e6i)T + (6.00e15 + 5.03e15i)T^{2} \) |
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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
−13.83735385121327424846906931651, −12.15708807259068298195141822143, −10.87493779398124480459047312019, −10.26951747329798352242367820613, −9.074952684679517980143291689658, −7.85300577542194374953030839825, −5.96608147630969281001313196392, −4.08883339474507246092321251623, −3.07669967887439319524130291682, −0.831071189573786450087928275666,
0.997594772795874764688210137680, 2.49472573222536393438904515181, 5.08607813982018902978395478331, 6.45931576488427194385826825772, 7.49431987106399448511695283294, 8.768854702782200426771553747121, 9.731454597532871776431529518832, 11.65571880064961694204317398388, 12.49228673332682350122385965156, 13.72957255512336127780697006354
