| L(s) = 1 | + (−1.58 + 0.575i)2-s + (−0.564 + 3.20i)3-s + (0.639 − 0.536i)4-s + (−0.766 − 0.642i)5-s + (−0.950 − 5.39i)6-s + (0.274 + 0.474i)7-s + (0.981 − 1.69i)8-s + (−7.11 − 2.59i)9-s + (1.58 + 0.575i)10-s + (−0.165 + 0.286i)11-s + (1.35 + 2.35i)12-s + (0.837 + 4.74i)13-s + (−0.707 − 0.593i)14-s + (2.49 − 2.09i)15-s + (−0.863 + 4.89i)16-s + (−4.96 + 1.80i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 + 0.407i)2-s + (−0.326 + 1.84i)3-s + (0.319 − 0.268i)4-s + (−0.342 − 0.287i)5-s + (−0.388 − 2.20i)6-s + (0.103 + 0.179i)7-s + (0.346 − 0.600i)8-s + (−2.37 − 0.863i)9-s + (0.500 + 0.182i)10-s + (−0.0499 + 0.0864i)11-s + (0.391 + 0.678i)12-s + (0.232 + 1.31i)13-s + (−0.189 − 0.158i)14-s + (0.643 − 0.539i)15-s + (−0.215 + 1.22i)16-s + (−1.20 + 0.438i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00369264 - 0.391668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00369264 - 0.391668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-4.30 + 0.670i)T \) |
| good | 2 | \( 1 + (1.58 - 0.575i)T + (1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (0.564 - 3.20i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-0.274 - 0.474i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.165 - 0.286i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.837 - 4.74i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (4.96 - 1.80i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.850 - 0.713i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.01 - 1.09i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.01 - 5.21i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.67T + 37T^{2} \) |
| 41 | \( 1 + (1.37 - 7.79i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.25 - 1.05i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.32 - 1.57i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-5.15 + 4.32i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-6.39 + 2.32i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.520 + 0.436i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-7.30 - 2.65i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (0.832 + 0.698i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.42 + 13.7i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.243 - 1.38i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.427 - 0.740i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.52 - 14.3i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.59 + 2.40i)T + (74.3 - 62.3i)T^{2} \) |
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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
−15.08047548146317085166998486578, −13.77072497979580082458680321344, −11.90810456519367387786840812599, −10.95471656653267707776098570656, −9.955682399783838081403285057264, −9.079327603653561009100447584708, −8.469120920798747154004793673755, −6.65661133726165779866884823031, −4.91336157761132813331137916218, −3.86707814077619721716954688773,
0.71116334273059493876785043085, 2.48365328350370502895107669411, 5.56541766354752949247030836104, 7.05292091070601609836195954392, 7.86474847012016161566909370441, 8.729654010876811538185071388367, 10.42492197887713010637675739395, 11.34234267033434442638986792803, 12.17202679396286379829872013252, 13.39741100508404190705144702861
