| L(s) = 1 | − 2.14i·3-s + (1.82 − 1.29i)5-s − 2.12·7-s − 1.61·9-s + 0.796i·11-s + (3.57 − 0.479i)13-s + (−2.77 − 3.92i)15-s − 4.56i·17-s − 0.220i·19-s + 4.56i·21-s − 4.73i·23-s + (1.65 − 4.71i)25-s − 2.96i·27-s − 8.17·29-s − 1.17i·31-s + ⋯ |
| L(s) = 1 | − 1.24i·3-s + (0.815 − 0.578i)5-s − 0.801·7-s − 0.539·9-s + 0.240i·11-s + (0.991 − 0.132i)13-s + (−0.717 − 1.01i)15-s − 1.10i·17-s − 0.0504i·19-s + 0.995i·21-s − 0.987i·23-s + (0.331 − 0.943i)25-s − 0.571i·27-s − 1.51·29-s − 0.211i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593117095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593117095\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.82 + 1.29i)T \) |
| 13 | \( 1 + (-3.57 + 0.479i)T \) |
| good | 3 | \( 1 + 2.14iT - 3T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 0.796iT - 11T^{2} \) |
| 17 | \( 1 + 4.56iT - 17T^{2} \) |
| 19 | \( 1 + 0.220iT - 19T^{2} \) |
| 23 | \( 1 + 4.73iT - 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 0.121T + 37T^{2} \) |
| 41 | \( 1 - 4.87iT - 41T^{2} \) |
| 43 | \( 1 + 3.16iT - 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 3.72iT - 59T^{2} \) |
| 61 | \( 1 + 3.36T + 61T^{2} \) |
| 67 | \( 1 + 7.43T + 67T^{2} \) |
| 71 | \( 1 - 15.2iT - 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 - 9.05T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 15.6iT - 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−9.463193849489205524185818696488, −8.841408358115118665201010317547, −7.85189512480291885949757750735, −6.99112085174587889219270130628, −6.27820605509753644838761551790, −5.60855149171076563510365172246, −4.36598013193278022791907065587, −2.92889627297938234199895610417, −1.87730707988957579848251393727, −0.72559926813134668166592820908,
1.82904635834694899251108962025, 3.44317526548194928790469740436, 3.69543269382130856030006248910, 5.13291180075834262900117142949, 5.95808212867716784994116915845, 6.60075098895442071526568903942, 7.80627310159459058643077832221, 9.076434810464180759901448218986, 9.392548812032858891213040425260, 10.28010435452693779252678468184
