| L(s) = 1 | − 4.35i·3-s + (−2.5 − 10.8i)5-s − 8.71i·7-s + 7.99·9-s + 11·11-s + 69.7i·13-s + (−47.5 + 10.8i)15-s + 26.1i·17-s − 68·19-s − 38.0·21-s − 117. i·23-s + (−112. + 54.4i)25-s − 152. i·27-s − 260·29-s − 175·31-s + ⋯ |
| L(s) = 1 | − 0.838i·3-s + (−0.223 − 0.974i)5-s − 0.470i·7-s + 0.296·9-s + 0.301·11-s + 1.48i·13-s + (−0.817 + 0.187i)15-s + 0.373i·17-s − 0.821·19-s − 0.394·21-s − 1.06i·23-s + (−0.900 + 0.435i)25-s − 1.08i·27-s − 1.66·29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.223 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.5 + 10.8i)T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 4.35iT - 27T^{2} \) |
| 7 | \( 1 + 8.71iT - 343T^{2} \) |
| 13 | \( 1 - 69.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 26.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 68T + 6.85e3T^{2} \) |
| 23 | \( 1 + 117. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 260T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175T + 2.97e4T^{2} \) |
| 37 | \( 1 - 169. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 380T + 6.89e4T^{2} \) |
| 43 | \( 1 + 305. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 305. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 453. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 143T + 2.05e5T^{2} \) |
| 61 | \( 1 - 676T + 2.26e5T^{2} \) |
| 67 | \( 1 - 527. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 218T + 4.93e5T^{2} \) |
| 83 | \( 1 - 758. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 771. iT - 9.12e5T^{2} \) |
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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
−8.973524909901191267283101494660, −8.368820472229602627060790169402, −7.29532797248977256757604664800, −6.78587756900720496850875367245, −5.74094225630581476089605988780, −4.42910729672148912344157577890, −3.95996554141189102432383437016, −2.03778139979098331335060036385, −1.32368266717115646098174359939, 0,
1.96143407284494698426627859229, 3.28368367897313935058429439411, 3.82825849100502005167608208822, 5.14681888173533660295886295544, 5.87160098364068095329608726569, 7.05289686269306588950857296615, 7.72136227794565121108694769649, 8.820366721960896394006387520889, 9.685802593369645885903864558656
