| L(s) = 1 | − 2.09·2-s − 3-s + 2.39·4-s + 1.55·5-s + 2.09·6-s − 4.13·7-s − 0.819·8-s + 9-s − 3.26·10-s − 4.43·11-s − 2.39·12-s + 3.67·13-s + 8.66·14-s − 1.55·15-s − 3.06·16-s − 2.09·18-s + 0.406·19-s + 3.72·20-s + 4.13·21-s + 9.28·22-s + 6.47·23-s + 0.819·24-s − 2.57·25-s − 7.70·26-s − 27-s − 9.88·28-s + 4.48·29-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.577·3-s + 1.19·4-s + 0.696·5-s + 0.855·6-s − 1.56·7-s − 0.289·8-s + 0.333·9-s − 1.03·10-s − 1.33·11-s − 0.690·12-s + 1.02·13-s + 2.31·14-s − 0.402·15-s − 0.766·16-s − 0.493·18-s + 0.0932·19-s + 0.833·20-s + 0.902·21-s + 1.97·22-s + 1.35·23-s + 0.167·24-s − 0.514·25-s − 1.51·26-s − 0.192·27-s − 1.86·28-s + 0.833·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4489618893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4489618893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 19 | \( 1 - 0.406T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 - 4.48T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 0.713T + 37T^{2} \) |
| 41 | \( 1 - 2.00T + 41T^{2} \) |
| 43 | \( 1 + 4.13T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 - 6.11T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 - 6.53T + 67T^{2} \) |
| 71 | \( 1 - 1.63T + 71T^{2} \) |
| 73 | \( 1 - 4.72T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 9.33T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−10.11786246977877446486430880745, −9.378571144117197236415680214663, −8.767337493175540453767909076824, −7.64767854723817364192466307117, −6.83477424474087029569513266380, −6.08098937429479341916426991176, −5.17120298174708042964330381471, −3.47258572652382249659463191119, −2.21550498546091113223448641643, −0.67093240667363861203296822056,
0.67093240667363861203296822056, 2.21550498546091113223448641643, 3.47258572652382249659463191119, 5.17120298174708042964330381471, 6.08098937429479341916426991176, 6.83477424474087029569513266380, 7.64767854723817364192466307117, 8.767337493175540453767909076824, 9.378571144117197236415680214663, 10.11786246977877446486430880745
