| L(s) = 1 | + 2-s − 3-s + 4-s + 1.69·5-s − 6-s − 0.605·7-s + 8-s + 9-s + 1.69·10-s + 5.02·11-s − 12-s − 2.72·13-s − 0.605·14-s − 1.69·15-s + 16-s − 5.51·17-s + 18-s − 19-s + 1.69·20-s + 0.605·21-s + 5.02·22-s − 7.58·23-s − 24-s − 2.14·25-s − 2.72·26-s − 27-s − 0.605·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.755·5-s − 0.408·6-s − 0.228·7-s + 0.353·8-s + 0.333·9-s + 0.534·10-s + 1.51·11-s − 0.288·12-s − 0.755·13-s − 0.161·14-s − 0.436·15-s + 0.250·16-s − 1.33·17-s + 0.235·18-s − 0.229·19-s + 0.377·20-s + 0.132·21-s + 1.07·22-s − 1.58·23-s − 0.204·24-s − 0.428·25-s − 0.533·26-s − 0.192·27-s − 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 1.69T + 5T^{2} \) |
| 7 | \( 1 + 0.605T + 7T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 23 | \( 1 + 7.58T + 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 - 0.396T + 41T^{2} \) |
| 43 | \( 1 + 3.81T + 43T^{2} \) |
| 47 | \( 1 - 4.86T + 47T^{2} \) |
| 59 | \( 1 + 1.33T + 59T^{2} \) |
| 61 | \( 1 - 9.23T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 1.45T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 6.47T + 97T^{2} \) |
| 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
−7.29668411889142586794555815655, −6.87059184828591898726118608231, −6.10742624483288885342340073140, −5.77517458500292846683011136421, −4.77185919909646282146728881732, −4.17974797655165339745948462469, −3.42411970317636698105628111503, −2.12384199318458520680332772520, −1.68237932739504729314317416924, 0,
1.68237932739504729314317416924, 2.12384199318458520680332772520, 3.42411970317636698105628111503, 4.17974797655165339745948462469, 4.77185919909646282146728881732, 5.77517458500292846683011136421, 6.10742624483288885342340073140, 6.87059184828591898726118608231, 7.29668411889142586794555815655
