L(s) = 1 | + 0.186·3-s + 1.67·5-s + 0.961·7-s − 2.96·9-s − 4.26·11-s + 2.35·13-s + 0.312·15-s + 4.37·17-s + 6.62·19-s + 0.179·21-s − 8.68·23-s − 2.19·25-s − 1.11·27-s + 7.12·29-s − 5.75·31-s − 0.797·33-s + 1.60·35-s − 5.95·37-s + 0.439·39-s − 3.99·41-s + 4.31·43-s − 4.96·45-s − 13.2·47-s − 6.07·49-s + 0.818·51-s − 1.43·53-s − 7.14·55-s + ⋯ |
L(s) = 1 | + 0.107·3-s + 0.748·5-s + 0.363·7-s − 0.988·9-s − 1.28·11-s + 0.651·13-s + 0.0807·15-s + 1.06·17-s + 1.51·19-s + 0.0392·21-s − 1.81·23-s − 0.439·25-s − 0.214·27-s + 1.32·29-s − 1.03·31-s − 0.138·33-s + 0.272·35-s − 0.978·37-s + 0.0703·39-s − 0.624·41-s + 0.657·43-s − 0.739·45-s − 1.93·47-s − 0.867·49-s + 0.114·51-s − 0.197·53-s − 0.963·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
good | 3 | \( 1 - 0.186T + 3T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 - 0.961T + 7T^{2} \) |
| 11 | \( 1 + 4.26T + 11T^{2} \) |
| 13 | \( 1 - 2.35T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 + 5.95T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 4.31T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 + 1.43T + 53T^{2} \) |
| 59 | \( 1 - 0.806T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 6.51T + 79T^{2} \) |
| 83 | \( 1 + 0.155T + 83T^{2} \) |
| 89 | \( 1 + 6.97T + 89T^{2} \) |
| 97 | \( 1 - 0.187T + 97T^{2} \) |
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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
−7.69706174448024952348298176921, −6.63110976428692944894389970069, −5.79710953443416168570561429736, −5.49338082110439888757946649329, −4.88805811431397725850176641116, −3.62522999235540791781755812298, −3.08053191014165698678869041000, −2.20056001072076181550592630112, −1.37731063820308666152933916690, 0,
1.37731063820308666152933916690, 2.20056001072076181550592630112, 3.08053191014165698678869041000, 3.62522999235540791781755812298, 4.88805811431397725850176641116, 5.49338082110439888757946649329, 5.79710953443416168570561429736, 6.63110976428692944894389970069, 7.69706174448024952348298176921