| L(s) = 1 | − 2.59·2-s − 1.64·3-s + 4.72·4-s − 1.77·5-s + 4.26·6-s − 2.35·7-s − 7.07·8-s − 0.299·9-s + 4.59·10-s + 11-s − 7.77·12-s − 1.76·13-s + 6.10·14-s + 2.91·15-s + 8.90·16-s − 17-s + 0.776·18-s + 4.33·19-s − 8.37·20-s + 3.86·21-s − 2.59·22-s − 1.79·23-s + 11.6·24-s − 1.86·25-s + 4.58·26-s + 5.42·27-s − 11.1·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.948·3-s + 2.36·4-s − 0.791·5-s + 1.74·6-s − 0.889·7-s − 2.50·8-s − 0.0997·9-s + 1.45·10-s + 0.301·11-s − 2.24·12-s − 0.490·13-s + 1.63·14-s + 0.751·15-s + 2.22·16-s − 0.242·17-s + 0.182·18-s + 0.993·19-s − 1.87·20-s + 0.844·21-s − 0.553·22-s − 0.375·23-s + 2.37·24-s − 0.372·25-s + 0.899·26-s + 1.04·27-s − 2.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1974182067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1974182067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 + 1.64T + 3T^{2} \) |
| 5 | \( 1 + 1.77T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 19 | \( 1 - 4.33T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 5.71T + 29T^{2} \) |
| 31 | \( 1 - 0.869T + 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 - 8.84T + 41T^{2} \) |
| 47 | \( 1 + 5.13T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 - 0.969T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 + 4.15T + 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 + 9.63T + 73T^{2} \) |
| 79 | \( 1 + 9.32T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 2.23T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−7.79070257809209966662938315543, −7.34289278843004598455839530969, −6.55268725050580042333873946585, −6.18954493882715514564468421031, −5.31118704189599669087085599819, −4.24832281836750799595912324243, −3.18369698694229518621392555714, −2.49893841722560349324444946769, −1.18403054820303505340267348790, −0.34952845065855581605334346895,
0.34952845065855581605334346895, 1.18403054820303505340267348790, 2.49893841722560349324444946769, 3.18369698694229518621392555714, 4.24832281836750799595912324243, 5.31118704189599669087085599819, 6.18954493882715514564468421031, 6.55268725050580042333873946585, 7.34289278843004598455839530969, 7.79070257809209966662938315543
