| 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
−14.19474052303657881734649973769, −13.75484490241675565346458321462, −11.58116394022725809368726737695, −10.70310169467002818500583691553, −9.578007301534968190170489211829, −8.493805353611764407183383847956, −7.48904566283470879952686965421, −6.61246647881494741278960431942, −4.59687213645666407739942099320, −3.21704464367050128272269778652,
1.34884159747390622706450761636, 3.44504036713054363022676650627, 4.62716700224192873427071052279, 7.22704171604021889618871511371, 8.248978180773246420501775958661, 8.880858593591877289921323225525, 9.942988814696728204773355367859, 11.72690031572372002922453470441, 12.22175084410167264457715268330, 13.01474866376191266758435645971
