L(s) = 1 | + (2.81 + 0.285i)2-s + (7.83 + 1.60i)4-s + (14.4 − 8.36i)5-s + (−16.7 − 9.65i)7-s + (21.5 + 6.74i)8-s + (43.1 − 19.4i)10-s + (−2.44 + 4.22i)11-s + (6.03 + 10.4i)13-s + (−44.2 − 31.9i)14-s + (58.8 + 25.1i)16-s + 71.2i·17-s − 68.3i·19-s + (127. − 42.3i)20-s + (−8.07 + 11.2i)22-s + (68.0 + 117. i)23-s + ⋯ |
L(s) = 1 | + (0.994 + 0.100i)2-s + (0.979 + 0.200i)4-s + (1.29 − 0.748i)5-s + (−0.902 − 0.521i)7-s + (0.954 + 0.298i)8-s + (1.36 − 0.613i)10-s + (−0.0669 + 0.115i)11-s + (0.128 + 0.223i)13-s + (−0.845 − 0.609i)14-s + (0.919 + 0.392i)16-s + 1.01i·17-s − 0.824i·19-s + (1.41 − 0.473i)20-s + (−0.0782 + 0.108i)22-s + (0.616 + 1.06i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.16202 - 0.343496i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.16202 - 0.343496i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.81 - 0.285i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-14.4 + 8.36i)T + (62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (16.7 + 9.65i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (2.44 - 4.22i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-6.03 - 10.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 71.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 68.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-68.0 - 117. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (190. + 109. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (285. - 164. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-29.5 + 17.0i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (0.558 + 0.322i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (93.4 - 161. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 266. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-104. - 180. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-0.801 + 1.38i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-371. + 214. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-68.5 - 39.5i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-462. + 801. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.04e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-733. + 1.26e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.10592010905352237679836256259, −12.75101999164878670097700521026, −11.18471684126352424868795391490, −10.05001718693125450874257381712, −9.012921025679973797780724156810, −7.23061134994590304564055111941, −6.11359779391283238484857972975, −5.14954471095484840881754337212, −3.59439699503097070219424423668, −1.78071700580366098120759164201,
2.21093546168783796977701253188, 3.34815641854361311941406008113, 5.38417863194738593221030301230, 6.18496948205364087401136376363, 7.19569356433683287740339919101, 9.266679369397060511311588266005, 10.22733020178283534996481509809, 11.18727128590288978851651824927, 12.57384711931299123875446213663, 13.23371712212639422546308335360