| L(s) = 1 | − 1.57·2-s − 0.803·3-s + 0.480·4-s − 2.40·5-s + 1.26·6-s + 2.39·8-s − 2.35·9-s + 3.78·10-s − 2.78·11-s − 0.386·12-s + 2.64·13-s + 1.92·15-s − 4.73·16-s − 4.93·17-s + 3.70·18-s + 3.29·19-s − 1.15·20-s + 4.39·22-s + 6.57·23-s − 1.92·24-s + 0.765·25-s − 4.16·26-s + 4.30·27-s + 4.39·29-s − 3.03·30-s + 8.44·31-s + 2.66·32-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 0.463·3-s + 0.240·4-s − 1.07·5-s + 0.516·6-s + 0.846·8-s − 0.784·9-s + 1.19·10-s − 0.840·11-s − 0.111·12-s + 0.732·13-s + 0.498·15-s − 1.18·16-s − 1.19·17-s + 0.874·18-s + 0.754·19-s − 0.258·20-s + 0.936·22-s + 1.37·23-s − 0.392·24-s + 0.153·25-s − 0.815·26-s + 0.827·27-s + 0.815·29-s − 0.554·30-s + 1.51·31-s + 0.470·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 3 | \( 1 + 0.803T + 3T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 - 6.57T + 23T^{2} \) |
| 29 | \( 1 - 4.39T + 29T^{2} \) |
| 31 | \( 1 - 8.44T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 9.96T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 7.28T + 61T^{2} \) |
| 67 | \( 1 - 4.55T + 67T^{2} \) |
| 71 | \( 1 + 1.44T + 71T^{2} \) |
| 73 | \( 1 - 6.71T + 73T^{2} \) |
| 79 | \( 1 - 5.77T + 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 + 9.81T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.447419734167772802719717154045, −8.176441921257370297851082069457, −7.16063763358208409355987378640, −6.58873450891650284283017144373, −5.30195056208135455661678447029, −4.71704298733197349999698910148, −3.62975033850634045605060025180, −2.57054025380865118539027737092, −0.986440893895717128684204469697, 0,
0.986440893895717128684204469697, 2.57054025380865118539027737092, 3.62975033850634045605060025180, 4.71704298733197349999698910148, 5.30195056208135455661678447029, 6.58873450891650284283017144373, 7.16063763358208409355987378640, 8.176441921257370297851082069457, 8.447419734167772802719717154045
