| L(s) = 1 | + (2 − 3.46i)2-s + (4.5 + 7.79i)3-s + (−7.99 − 13.8i)4-s + (−43 + 74.4i)5-s + 36·6-s + (24.5 + 127. i)7-s − 63.9·8-s + (−40.5 + 70.1i)9-s + (172 + 297. i)10-s + (−17 − 29.4i)11-s + (72 − 124. i)12-s − 3·13-s + (490 + 169. i)14-s − 774.·15-s + (−128 + 221. i)16-s + (952 + 1.64e3i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.769 + 1.33i)5-s + 0.408·6-s + (0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.543 + 0.942i)10-s + (−0.0423 − 0.0733i)11-s + (0.144 − 0.249i)12-s − 0.00492·13-s + (0.668 + 0.231i)14-s − 0.888·15-s + (−0.125 + 0.216i)16-s + (0.798 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.27438 + 0.969495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27438 + 0.969495i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (-24.5 - 127. i)T \) |
| good | 5 | \( 1 + (43 - 74.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (17 + 29.4i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 3T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-952 - 1.64e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-744.5 + 1.28e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-112 + 193. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 6.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (865.5 + 1.49e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-3.81e3 + 6.61e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.54e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.84e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (9.23e3 - 1.59e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-9.97e3 - 1.72e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.59e4 - 2.75e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.88e4 + 4.99e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.02e4 - 5.24e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 4.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.04e4 + 1.80e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.52e4 + 2.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.10e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.94e4 + 5.10e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.19e5T + 8.58e9T^{2} \) |
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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
−14.92620898687039245536545767616, −14.47216167005077790726340839807, −12.75117476496957972854954856916, −11.44387136750234889762405014090, −10.71752267874604716279027041342, −9.284111729075777432914812719963, −7.71143991980332458817599347908, −5.81909624726626437707333953063, −3.87527031054882447815796470880, −2.60770790110591299546685996876,
0.802615122734954768208848093292, 3.85567495749267176241131341764, 5.25792909075009862551866829005, 7.31543270937134885296979933300, 8.056751827785019443898501584076, 9.464709440523692675298340368460, 11.59235423336549650335378403877, 12.62707534485783479501012057098, 13.59533718097096214751554718749, 14.65670869096059589267035278023
