| L(s) = 1 | + 1.08·2-s − 2.08·3-s − 0.817·4-s − 0.209·5-s − 2.26·6-s − 3.06·8-s + 1.35·9-s − 0.227·10-s + 6.03·11-s + 1.70·12-s − 3.67·13-s + 0.437·15-s − 1.69·16-s + 5.37·17-s + 1.47·18-s + 3.54·19-s + 0.171·20-s + 6.56·22-s − 1.30·23-s + 6.39·24-s − 4.95·25-s − 3.99·26-s + 3.43·27-s − 8.00·29-s + 0.475·30-s + 0.384·31-s + 4.28·32-s + ⋯ |
| L(s) = 1 | + 0.768·2-s − 1.20·3-s − 0.408·4-s − 0.0937·5-s − 0.926·6-s − 1.08·8-s + 0.452·9-s − 0.0720·10-s + 1.82·11-s + 0.492·12-s − 1.01·13-s + 0.112·15-s − 0.423·16-s + 1.30·17-s + 0.347·18-s + 0.812·19-s + 0.0383·20-s + 1.39·22-s − 0.271·23-s + 1.30·24-s − 0.991·25-s − 0.782·26-s + 0.660·27-s − 1.48·29-s + 0.0868·30-s + 0.0690·31-s + 0.757·32-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 | \( 1 - 1.08T + 2T^{2} \) |
| 3 | \( 1 + 2.08T + 3T^{2} \) |
| 5 | \( 1 + 0.209T + 5T^{2} \) |
| 11 | \( 1 - 6.03T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 - 3.54T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 - 0.384T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 43 | \( 1 - 0.824T + 43T^{2} \) |
| 47 | \( 1 + 5.11T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 7.77T + 73T^{2} \) |
| 79 | \( 1 + 6.04T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 0.520T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
| 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
−9.001140029809659485794622359314, −7.78412429633665032311926291225, −6.95951543402134946435272789148, −5.99152865787834151278956160709, −5.62593883081304205267153379519, −4.77608002990278672198734818311, −3.97907488030448197110278937260, −3.17200756969222704890183264599, −1.40197895365906947914838220522, 0,
1.40197895365906947914838220522, 3.17200756969222704890183264599, 3.97907488030448197110278937260, 4.77608002990278672198734818311, 5.62593883081304205267153379519, 5.99152865787834151278956160709, 6.95951543402134946435272789148, 7.78412429633665032311926291225, 9.001140029809659485794622359314
