| L(s) = 1 | − 0.979·2-s − 1.94·3-s − 1.03·4-s − 1.73·5-s + 1.90·6-s + 2.97·8-s + 0.798·9-s + 1.70·10-s − 0.677·11-s + 2.02·12-s − 1.88·13-s + 3.39·15-s − 0.839·16-s − 3.26·17-s − 0.782·18-s − 1.13·19-s + 1.80·20-s + 0.663·22-s + 5.51·23-s − 5.80·24-s − 1.97·25-s + 1.84·26-s + 4.29·27-s + 4.66·29-s − 3.32·30-s + 5.77·31-s − 5.13·32-s + ⋯ |
| L(s) = 1 | − 0.692·2-s − 1.12·3-s − 0.519·4-s − 0.777·5-s + 0.779·6-s + 1.05·8-s + 0.266·9-s + 0.539·10-s − 0.204·11-s + 0.585·12-s − 0.522·13-s + 0.875·15-s − 0.209·16-s − 0.790·17-s − 0.184·18-s − 0.260·19-s + 0.404·20-s + 0.141·22-s + 1.14·23-s − 1.18·24-s − 0.394·25-s + 0.362·26-s + 0.825·27-s + 0.866·29-s − 0.606·30-s + 1.03·31-s − 0.907·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.979T + 2T^{2} \) |
| 3 | \( 1 + 1.94T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 + 0.677T + 11T^{2} \) |
| 13 | \( 1 + 1.88T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 - 4.66T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 1.55T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 61 | \( 1 + 0.260T + 61T^{2} \) |
| 67 | \( 1 - 0.416T + 67T^{2} \) |
| 71 | \( 1 - 6.70T + 71T^{2} \) |
| 73 | \( 1 - 3.71T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.708514994185674158093190100260, −8.096126149825712960854362120367, −7.25041868135159129809566710061, −6.52159374844989472348837038861, −5.46193997177433351029894404250, −4.71154558184589933208384849404, −4.10988376400506168228863140832, −2.66707223560755616415831962389, −0.972879438783941471743479793051, 0,
0.972879438783941471743479793051, 2.66707223560755616415831962389, 4.10988376400506168228863140832, 4.71154558184589933208384849404, 5.46193997177433351029894404250, 6.52159374844989472348837038861, 7.25041868135159129809566710061, 8.096126149825712960854362120367, 8.708514994185674158093190100260
