| L(s) = 1 | + 2.98i·3-s + (0.274 − 2.21i)5-s + 0.980i·7-s − 5.92·9-s − 6.14·11-s − 6.37i·13-s + (6.62 + 0.819i)15-s − 3.36i·17-s − 1.08·19-s − 2.92·21-s − i·23-s + (−4.84 − 1.21i)25-s − 8.72i·27-s + 0.271·29-s − 8.77·31-s + ⋯ |
| L(s) = 1 | + 1.72i·3-s + (0.122 − 0.992i)5-s + 0.370i·7-s − 1.97·9-s − 1.85·11-s − 1.76i·13-s + (1.71 + 0.211i)15-s − 0.815i·17-s − 0.248·19-s − 0.639·21-s − 0.208i·23-s + (−0.969 − 0.243i)25-s − 1.68i·27-s + 0.0503·29-s − 1.57·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.235683 - 0.266607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.235683 - 0.266607i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.274 + 2.21i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 2.98iT - 3T^{2} \) |
| 7 | \( 1 - 0.980iT - 7T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 13 | \( 1 + 6.37iT - 13T^{2} \) |
| 17 | \( 1 + 3.36iT - 17T^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 29 | \( 1 - 0.271T + 29T^{2} \) |
| 31 | \( 1 + 8.77T + 31T^{2} \) |
| 37 | \( 1 - 8.84iT - 37T^{2} \) |
| 41 | \( 1 + 4.85T + 41T^{2} \) |
| 43 | \( 1 - 1.87iT - 43T^{2} \) |
| 47 | \( 1 + 0.196iT - 47T^{2} \) |
| 53 | \( 1 + 1.93iT - 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 6.00T + 61T^{2} \) |
| 67 | \( 1 + 2.26iT - 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 1.38iT - 73T^{2} \) |
| 79 | \( 1 - 4.67T + 79T^{2} \) |
| 83 | \( 1 + 15.7iT - 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 10.2iT - 97T^{2} \) |
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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
−9.970807371580224637557565774509, −9.136490963317054397196257177666, −8.352878586918599907401927934718, −7.73270401210841523592814931927, −5.82928440123939802807979491528, −5.13564677853368323746714267749, −4.88543862405469689592327310588, −3.48758784072080185789513968927, −2.61966299320087170937948563312, −0.15286113413863810483829412898,
1.84160966702817286422489081635, 2.45187992980964674555240734534, 3.78886608209461164514487223154, 5.40739434853911359118135482638, 6.27264951684507059698487470296, 7.06444003014262483233471476136, 7.50534772050590849472192090301, 8.308995516043885261282630332802, 9.390317159894221481490555989578, 10.67503769696353576117340482345
