| L(s) = 1 | + (−0.684 − 1.23i)2-s + (1.44 − 0.948i)3-s + (−1.06 + 1.69i)4-s + (−2.41 − 2.87i)5-s + (−2.16 − 1.14i)6-s + (0.0801 + 0.220i)7-s + (2.82 + 0.153i)8-s + (1.20 − 2.74i)9-s + (−1.90 + 4.95i)10-s + (2.36 + 1.98i)11-s + (0.0674 + 3.46i)12-s + (−0.155 − 0.880i)13-s + (0.217 − 0.250i)14-s + (−6.21 − 1.87i)15-s + (−1.74 − 3.60i)16-s + (−2.18 + 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.484 − 0.874i)2-s + (0.836 − 0.547i)3-s + (−0.531 + 0.847i)4-s + (−1.07 − 1.28i)5-s + (−0.884 − 0.467i)6-s + (0.0303 + 0.0832i)7-s + (0.998 + 0.0544i)8-s + (0.400 − 0.916i)9-s + (−0.602 + 1.56i)10-s + (0.712 + 0.597i)11-s + (0.0194 + 0.999i)12-s + (−0.0430 − 0.244i)13-s + (0.0581 − 0.0668i)14-s + (−1.60 − 0.484i)15-s + (−0.435 − 0.900i)16-s + (−0.530 + 0.306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.472533 - 0.753699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472533 - 0.753699i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.684 + 1.23i)T \) |
| 3 | \( 1 + (-1.44 + 0.948i)T \) |
| good | 5 | \( 1 + (2.41 + 2.87i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.0801 - 0.220i)T + (-5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-2.36 - 1.98i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.155 + 0.880i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.18 - 1.26i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.71 - 3.30i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.764 - 0.278i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.40 - 1.48i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.710 - 1.95i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (0.697 + 1.20i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.54 - 1.50i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.07 + 2.47i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (10.1 - 3.69i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 4.76iT - 53T^{2} \) |
| 59 | \( 1 + (7.52 - 6.31i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (5.15 - 1.87i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.13 - 0.200i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.28 - 7.42i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.27 - 2.21i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.54 + 1.50i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.675 + 3.83i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (0.286 + 0.165i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.1 + 8.52i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.01474866376191266758435645971, −12.22175084410167264457715268330, −11.72690031572372002922453470441, −9.942988814696728204773355367859, −8.880858593591877289921323225525, −8.248978180773246420501775958661, −7.22704171604021889618871511371, −4.62716700224192873427071052279, −3.44504036713054363022676650627, −1.34884159747390622706450761636,
3.21704464367050128272269778652, 4.59687213645666407739942099320, 6.61246647881494741278960431942, 7.48904566283470879952686965421, 8.493805353611764407183383847956, 9.578007301534968190170489211829, 10.70310169467002818500583691553, 11.58116394022725809368726737695, 13.75484490241675565346458321462, 14.19474052303657881734649973769
