| L(s) = 1 | + (−3.81 − 1.49i)2-s + (−2.12 − 4.74i)3-s + (6.41 + 5.95i)4-s + (16.4 + 11.1i)5-s + (1.01 + 21.2i)6-s + (−10.6 − 15.1i)7-s + (−1.34 − 2.78i)8-s + (−17.9 + 20.1i)9-s + (−45.8 − 67.1i)10-s + (−7.93 − 52.6i)11-s + (14.5 − 43.0i)12-s + (−50.8 + 40.5i)13-s + (17.9 + 73.6i)14-s + (18.1 − 101. i)15-s + (−4.28 − 57.2i)16-s + (−38.2 − 11.8i)17-s + ⋯ |
| L(s) = 1 | + (−1.34 − 0.528i)2-s + (−0.409 − 0.912i)3-s + (0.802 + 0.744i)4-s + (1.46 + 1.00i)5-s + (0.0693 + 1.44i)6-s + (−0.574 − 0.818i)7-s + (−0.0592 − 0.123i)8-s + (−0.664 + 0.747i)9-s + (−1.44 − 2.12i)10-s + (−0.217 − 1.44i)11-s + (0.350 − 1.03i)12-s + (−1.08 + 0.865i)13-s + (0.341 + 1.40i)14-s + (0.311 − 1.74i)15-s + (−0.0670 − 0.894i)16-s + (−0.545 − 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0101537 + 0.0145742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0101537 + 0.0145742i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.12 + 4.74i)T \) |
| 7 | \( 1 + (10.6 + 15.1i)T \) |
| good | 2 | \( 1 + (3.81 + 1.49i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-16.4 - 11.1i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (7.93 + 52.6i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (50.8 - 40.5i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (38.2 + 11.8i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (44.3 - 25.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-22.6 - 73.4i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (77.9 - 17.8i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (241. + 139. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (93.7 - 87.0i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-162. + 78.0i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (40.4 + 19.4i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (151. - 385. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (12.7 - 13.7i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-25.1 + 17.1i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (127. + 137. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (270. - 469. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-152. - 34.8i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-150. + 58.9i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (283. + 490. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (526. - 660. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (1.02e3 + 154. i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + 626. iT - 9.12e5T^{2} \) |
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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
−12.92387829439956026181190261745, −11.30551350606280989716364226353, −10.85431005059274186436935345444, −9.883865289311764738660091875047, −9.042277389765377839506417951555, −7.54371665243758928106750987765, −6.73713760461728844122281407946, −5.68060867807566802682222883232, −2.79574123502064820033532695866, −1.68430715248846713868861578582,
0.01317296606415028505920565293, 2.14647134079504604143678310425, 4.79634460534921412474144192255, 5.71978340750525630495300780508, 6.89623299417935674769927923161, 8.597993496009242526192259668506, 9.320534934994734000057913889064, 9.845590898060199105323492442333, 10.55677504138858898826668603116, 12.48321437932674672276395572792
