| L(s) = 1 | + (−2.59 − 1.11i)2-s + (0.142 + 0.390i)3-s + (5.51 + 5.79i)4-s + (−20.2 − 3.57i)5-s + (0.0661 − 1.17i)6-s + (−16.1 + 9.30i)7-s + (−7.86 − 21.2i)8-s + (20.5 − 17.2i)9-s + (48.7 + 31.9i)10-s + (4.07 − 7.05i)11-s + (−1.48 + 2.98i)12-s + (26.9 + 9.81i)13-s + (52.2 − 6.20i)14-s + (−1.48 − 8.43i)15-s + (−3.23 + 63.9i)16-s + (32.2 + 27.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.918 − 0.394i)2-s + (0.0273 + 0.0752i)3-s + (0.689 + 0.724i)4-s + (−1.81 − 0.319i)5-s + (0.00450 − 0.0799i)6-s + (−0.870 + 0.502i)7-s + (−0.347 − 0.937i)8-s + (0.761 − 0.638i)9-s + (1.54 + 1.00i)10-s + (0.111 − 0.193i)11-s + (−0.0356 + 0.0716i)12-s + (0.575 + 0.209i)13-s + (0.997 − 0.118i)14-s + (−0.0256 − 0.145i)15-s + (−0.0504 + 0.998i)16-s + (0.460 + 0.386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0568i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.678173 - 0.0192948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678173 - 0.0192948i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.59 + 1.11i)T \) |
| 19 | \( 1 + (57.1 - 59.9i)T \) |
| good | 3 | \( 1 + (-0.142 - 0.390i)T + (-20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (20.2 + 3.57i)T + (117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (16.1 - 9.30i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-4.07 + 7.05i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-26.9 - 9.81i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-32.2 - 27.0i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-179. + 31.7i)T + (1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-15.3 + 12.9i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-8.46 - 14.6i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (110. + 303. i)T + (-5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-12.2 + 69.3i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-338. - 403. i)T + (-1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (82.4 + 467. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (45.7 - 54.5i)T + (-3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (709. - 125. i)T + (2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-430. - 513. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (127. - 724. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-407. + 148. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (-955. + 347. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (122. + 211. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (327. - 899. i)T + (-5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-741. + 884. i)T + (-1.58e5 - 8.98e5i)T^{2} \) |
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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
−12.47242120917577168078170943752, −11.48216026769049436067712767995, −10.54348832183299442774668961378, −9.245516757123247126939581987089, −8.523261885933464425184423989711, −7.45665174758246622047006495319, −6.43958173262325126729117684310, −4.10468480294144206484233567143, −3.25928993199813355347554086182, −0.842915964013817571501859472798,
0.69719634271811017733697512462, 3.15619552379525478994567715315, 4.63564643177794218575895333046, 6.66197984294798114187753080227, 7.32942152652500584967640448610, 8.133556119145631081531788294112, 9.356875701476139144005135480703, 10.62694958160685911841427725242, 11.16287137275724425904655330702, 12.36776790592187223279337529782
