L(s) = 1 | − 2-s − 3-s + 4-s − 2.03·5-s + 6-s − 3.87·7-s − 8-s + 9-s + 2.03·10-s − 2.68·11-s − 12-s + 5.11·13-s + 3.87·14-s + 2.03·15-s + 16-s − 17-s − 18-s + 0.117·19-s − 2.03·20-s + 3.87·21-s + 2.68·22-s − 1.09·23-s + 24-s − 0.843·25-s − 5.11·26-s − 27-s − 3.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.911·5-s + 0.408·6-s − 1.46·7-s − 0.353·8-s + 0.333·9-s + 0.644·10-s − 0.808·11-s − 0.288·12-s + 1.41·13-s + 1.03·14-s + 0.526·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.0268·19-s − 0.455·20-s + 0.846·21-s + 0.571·22-s − 0.228·23-s + 0.204·24-s − 0.168·25-s − 1.00·26-s − 0.192·27-s − 0.732·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 19 | \( 1 - 0.117T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 9.39T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 + 2.51T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 61 | \( 1 + 2.83T + 61T^{2} \) |
| 67 | \( 1 - 0.0798T + 67T^{2} \) |
| 71 | \( 1 - 9.94T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 6.86T + 97T^{2} \) |
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
−7.74489597322468892680056555858, −6.95714783601100004283950223091, −6.47144274492748692073445663352, −5.81435116886020793969094373916, −4.91392811787036826641545681975, −3.72364332128672985231008279648, −3.41361194716495021617717587015, −2.26597162129341100991695390934, −0.871009945638245149291169476026, 0,
0.871009945638245149291169476026, 2.26597162129341100991695390934, 3.41361194716495021617717587015, 3.72364332128672985231008279648, 4.91392811787036826641545681975, 5.81435116886020793969094373916, 6.47144274492748692073445663352, 6.95714783601100004283950223091, 7.74489597322468892680056555858