L(s) = 1 | + 2-s + 3-s + 4-s − 2.41·5-s + 6-s + 1.11·7-s + 8-s + 9-s − 2.41·10-s + 0.225·11-s + 12-s + 4.24·13-s + 1.11·14-s − 2.41·15-s + 16-s + 17-s + 18-s − 2.00·19-s − 2.41·20-s + 1.11·21-s + 0.225·22-s + 4.30·23-s + 24-s + 0.832·25-s + 4.24·26-s + 27-s + 1.11·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.08·5-s + 0.408·6-s + 0.419·7-s + 0.353·8-s + 0.333·9-s − 0.763·10-s + 0.0679·11-s + 0.288·12-s + 1.17·13-s + 0.296·14-s − 0.623·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.459·19-s − 0.540·20-s + 0.242·21-s + 0.0480·22-s + 0.897·23-s + 0.204·24-s + 0.166·25-s + 0.831·26-s + 0.192·27-s + 0.209·28-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
\(L(1)\) |
\(\approx\) |
\(3.779837149\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.779837149\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 - 0.225T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 19 | \( 1 + 2.00T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 + 0.556T + 29T^{2} \) |
| 31 | \( 1 - 8.00T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 + 2.49T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 - 9.35T + 47T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 8.08T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 0.775T + 73T^{2} \) |
| 79 | \( 1 + 4.12T + 79T^{2} \) |
| 83 | \( 1 - 7.37T + 83T^{2} \) |
| 89 | \( 1 - 5.90T + 89T^{2} \) |
| 97 | \( 1 + 4.80T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.199912308024037282621792187957, −7.35480616372110820455886086674, −6.72750097103749598541171009608, −5.94463372213433991402787692220, −4.97550210637571993394738989600, −4.33542240653498861864782952842, −3.63611111075307244435295020501, −3.10448882900171149402850600058, −1.99969325624139745872679479693, −0.928465286815895194914578884646,
0.928465286815895194914578884646, 1.99969325624139745872679479693, 3.10448882900171149402850600058, 3.63611111075307244435295020501, 4.33542240653498861864782952842, 4.97550210637571993394738989600, 5.94463372213433991402787692220, 6.72750097103749598541171009608, 7.35480616372110820455886086674, 8.199912308024037282621792187957