L(s) = 1 | − 2-s − 0.743·3-s + 4-s − 1.77·5-s + 0.743·6-s + 7-s − 8-s − 2.44·9-s + 1.77·10-s + 2.78·11-s − 0.743·12-s + 5.15·13-s − 14-s + 1.31·15-s + 16-s + 0.832·17-s + 2.44·18-s + 4.69·19-s − 1.77·20-s − 0.743·21-s − 2.78·22-s + 5.67·23-s + 0.743·24-s − 1.86·25-s − 5.15·26-s + 4.04·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.429·3-s + 0.5·4-s − 0.791·5-s + 0.303·6-s + 0.377·7-s − 0.353·8-s − 0.815·9-s + 0.559·10-s + 0.839·11-s − 0.214·12-s + 1.42·13-s − 0.267·14-s + 0.339·15-s + 0.250·16-s + 0.202·17-s + 0.576·18-s + 1.07·19-s − 0.395·20-s − 0.162·21-s − 0.593·22-s + 1.18·23-s + 0.151·24-s − 0.373·25-s − 1.01·26-s + 0.779·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.176007880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.176007880\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 + 0.743T + 3T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 - 0.832T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 - 5.67T + 23T^{2} \) |
| 29 | \( 1 - 5.31T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 3.10T + 37T^{2} \) |
| 41 | \( 1 + 0.224T + 41T^{2} \) |
| 43 | \( 1 - 1.94T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 6.26T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 6.18T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 0.347T + 73T^{2} \) |
| 79 | \( 1 + 9.73T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 6.60T + 89T^{2} \) |
| 97 | \( 1 + 2.27T + 97T^{2} \) |
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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
−8.301517349725183640146765671873, −7.30249893611618845908961335531, −6.90811167800755133080984127187, −5.87612511445875088367834631474, −5.49532993197726685943334281375, −4.34439009983680991710301145842, −3.57021245670830623301881207550, −2.84480531918772781585912136792, −1.44078064140553870537851288241, −0.71742269629833668869462023090,
0.71742269629833668869462023090, 1.44078064140553870537851288241, 2.84480531918772781585912136792, 3.57021245670830623301881207550, 4.34439009983680991710301145842, 5.49532993197726685943334281375, 5.87612511445875088367834631474, 6.90811167800755133080984127187, 7.30249893611618845908961335531, 8.301517349725183640146765671873