| L(s) = 1 | + (−0.678 − 1.24i)2-s + (−0.339 + 0.933i)3-s + (−1.07 + 1.68i)4-s + (0.0653 − 0.0115i)5-s + (1.38 − 0.212i)6-s + (1.72 − 2.98i)7-s + (2.82 + 0.194i)8-s + (1.54 + 1.29i)9-s + (−0.0586 − 0.0732i)10-s + (5.02 − 2.90i)11-s + (−1.20 − 1.57i)12-s + (0.741 + 2.03i)13-s + (−4.86 − 0.111i)14-s + (−0.0114 + 0.0648i)15-s + (−1.67 − 3.63i)16-s + (−2.63 + 2.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.480 − 0.877i)2-s + (−0.196 + 0.538i)3-s + (−0.539 + 0.842i)4-s + (0.0292 − 0.00515i)5-s + (0.566 − 0.0866i)6-s + (0.650 − 1.12i)7-s + (0.997 + 0.0686i)8-s + (0.514 + 0.431i)9-s + (−0.0185 − 0.0231i)10-s + (1.51 − 0.874i)11-s + (−0.348 − 0.455i)12-s + (0.205 + 0.564i)13-s + (−1.30 − 0.0298i)14-s + (−0.00295 + 0.0167i)15-s + (−0.418 − 0.908i)16-s + (−0.639 + 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.866930 - 0.302886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.866930 - 0.302886i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.678 + 1.24i)T \) |
| 19 | \( 1 + (1.75 - 3.99i)T \) |
| good | 3 | \( 1 + (0.339 - 0.933i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.0653 + 0.0115i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.72 + 2.98i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.02 + 2.90i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.741 - 2.03i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.63 - 2.21i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.43 + 8.16i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.84 + 3.38i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.29 - 3.97i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.627iT - 37T^{2} \) |
| 41 | \( 1 + (-7.06 - 2.57i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.788 - 0.139i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.03 + 3.38i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (5.64 + 0.995i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (4.20 + 5.00i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (6.93 + 1.22i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.861 - 1.02i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.65 - 9.40i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.81 - 0.658i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.51 + 0.551i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (8.56 + 4.94i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (12.0 - 4.38i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-9.04 + 7.58i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.73748275266235482340870849239, −11.48223519823504752468277985116, −10.87686359528537880734988529796, −10.08364915384325794742020882305, −8.934588267130524159509598077038, −7.951156828166449353594296590460, −6.56114101967345075946714742648, −4.44567902704213694612927687565, −3.89599108335564443959943091162, −1.52879215005194631855422663885,
1.65283308701118567079882223453, 4.43085953134134678618268439714, 5.77422711967290522115408297928, 6.77139074687861131098912285829, 7.69415701087561596975720154236, 9.061045044316047886004107741655, 9.519116699561573934104761186117, 11.20365787244806077128162377793, 12.08539797827606695368771972619, 13.16876396880245092306575282365
