| L(s) = 1 | + 2·2-s + (2.30 + 4.65i)3-s + 4·4-s + (−5.06 + 8.77i)5-s + (4.61 + 9.31i)6-s + (14.4 − 11.5i)7-s + 8·8-s + (−16.3 + 21.4i)9-s + (−10.1 + 17.5i)10-s + (19.0 + 33.0i)11-s + (9.23 + 18.6i)12-s + (−1.46 − 2.53i)13-s + (28.8 − 23.1i)14-s + (−52.5 − 3.32i)15-s + 16·16-s + (−56.0 + 97.0i)17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + (0.444 + 0.895i)3-s + 0.5·4-s + (−0.453 + 0.785i)5-s + (0.314 + 0.633i)6-s + (0.779 − 0.626i)7-s + 0.353·8-s + (−0.605 + 0.796i)9-s + (−0.320 + 0.555i)10-s + (0.522 + 0.904i)11-s + (0.222 + 0.447i)12-s + (−0.0311 − 0.0539i)13-s + (0.551 − 0.442i)14-s + (−0.904 − 0.0572i)15-s + 0.250·16-s + (−0.799 + 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.12510 + 1.71766i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12510 + 1.71766i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (-2.30 - 4.65i)T \) |
| 7 | \( 1 + (-14.4 + 11.5i)T \) |
| good | 5 | \( 1 + (5.06 - 8.77i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-19.0 - 33.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (1.46 + 2.53i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (56.0 - 97.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (39.2 + 68.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-73.5 + 127. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-137. + 238. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (25.1 + 43.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-1.16 - 2.01i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-184. + 318. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 399.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (50.8 - 88.0i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 27.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 612.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 624.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 514.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-590. + 1.02e3i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 340.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (312. - 540. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-349. - 604. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (345. - 599. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.38531909959618911973881962800, −12.00615102256227248919309436921, −10.82685091599925790252496530311, −10.48634038454108314987431667265, −8.848797618304693553014237167959, −7.64191687168079254846227617990, −6.48839995440703978264046138928, −4.62799923248275902019059571694, −4.02761209105613473510815136957, −2.41994715414349934713789381419,
1.25677990130994871382087673434, 2.95180391331499559523707992507, 4.61401088363771592724799902703, 5.87947189023992653741034385441, 7.19840263317027428863634316311, 8.381449461355983607791918750070, 9.051389340530947258695228023608, 11.18719710369460128131785716072, 11.90515623603996955836115845198, 12.63330140447982200407780502147
