| L(s) = 1 | − 2-s + 1.52·3-s + 4-s + 0.627·5-s − 1.52·6-s + 3.51·7-s − 8-s − 0.675·9-s − 0.627·10-s − 3.81·11-s + 1.52·12-s + 1.92·13-s − 3.51·14-s + 0.957·15-s + 16-s − 17-s + 0.675·18-s + 5.22·19-s + 0.627·20-s + 5.36·21-s + 3.81·22-s + 4.63·23-s − 1.52·24-s − 4.60·25-s − 1.92·26-s − 5.60·27-s + 3.51·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.880·3-s + 0.5·4-s + 0.280·5-s − 0.622·6-s + 1.33·7-s − 0.353·8-s − 0.225·9-s − 0.198·10-s − 1.15·11-s + 0.440·12-s + 0.534·13-s − 0.940·14-s + 0.247·15-s + 0.250·16-s − 0.242·17-s + 0.159·18-s + 1.19·19-s + 0.140·20-s + 1.17·21-s + 0.813·22-s + 0.965·23-s − 0.311·24-s − 0.921·25-s − 0.378·26-s − 1.07·27-s + 0.665·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.041001560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041001560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 - 0.627T + 5T^{2} \) |
| 7 | \( 1 - 3.51T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 - 1.92T + 13T^{2} \) |
| 19 | \( 1 - 5.22T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 + 4.45T + 29T^{2} \) |
| 31 | \( 1 - 9.42T + 31T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 - 11.9T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 1.12T + 71T^{2} \) |
| 73 | \( 1 - 3.24T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 8.90T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
| 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
−9.145421268209942471355904314751, −8.191307940447035213858898656572, −7.940272960077918342497835849167, −7.24709207883637339409909042542, −5.89478130586182477143545137548, −5.28292653031331438608805733474, −4.14866490718748042554986355782, −2.87708384379586808732899904746, −2.27164655278946769733780916926, −1.06983426880189464732777375447,
1.06983426880189464732777375447, 2.27164655278946769733780916926, 2.87708384379586808732899904746, 4.14866490718748042554986355782, 5.28292653031331438608805733474, 5.89478130586182477143545137548, 7.24709207883637339409909042542, 7.940272960077918342497835849167, 8.191307940447035213858898656572, 9.145421268209942471355904314751
