| L(s) = 1 | + (1.10 − 1.91i)5-s + (17.6 − 10.1i)11-s + 64.4i·13-s + (−63.9 − 110. i)17-s + (−7.57 − 4.37i)19-s + (−66.0 − 38.1i)23-s + (60.0 + 104. i)25-s + 190. i·29-s + (143. − 82.6i)31-s + (54.3 − 94.0i)37-s + 312.·41-s − 139.·43-s + (−223. + 387. i)47-s + (460. − 265. i)53-s − 45.0i·55-s + ⋯ |
| L(s) = 1 | + (0.0989 − 0.171i)5-s + (0.482 − 0.278i)11-s + 1.37i·13-s + (−0.912 − 1.58i)17-s + (−0.0915 − 0.0528i)19-s + (−0.598 − 0.345i)23-s + (0.480 + 0.832i)25-s + 1.22i·29-s + (0.828 − 0.478i)31-s + (0.241 − 0.418i)37-s + 1.19·41-s − 0.496·43-s + (−0.694 + 1.20i)47-s + (1.19 − 0.689i)53-s − 0.110i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.934240875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.934240875\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-1.10 + 1.91i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-17.6 + 10.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 64.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (63.9 + 110. i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (7.57 + 4.37i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (66.0 + 38.1i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 190. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-143. + 82.6i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-54.3 + 94.0i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 312.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 139.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (223. - 387. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-460. + 265. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-138. - 240. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (157. + 90.7i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-88.4 - 153. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 259. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (423. - 244. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (284. - 493. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-335. + 580. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 526. iT - 9.12e5T^{2} \) |
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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
−9.161845512117943079406621440730, −8.402581258056386729094029635038, −7.22103547936649140529294419832, −6.77429001189807762549566941255, −5.84940725066876305399624375935, −4.77898175750073349125128925373, −4.20264238313171420732069045720, −2.99311759948499422271070316782, −1.99391729078056608369308571536, −0.836067506688269267161734674127,
0.52154204556430233327137362048, 1.79779270357378446965413261690, 2.80176925260731193258479260415, 3.88416512976676922023142750539, 4.62484736407559953480442047599, 5.84430261111437045870702040404, 6.29692916010500775508819450815, 7.27713677455270189929313136068, 8.257187522878763712818625833913, 8.594346942048308288019182566405
