L(s) = 1 | − 2.62·3-s + 2.18·5-s + 1.99·7-s + 3.90·9-s + 13-s − 5.73·15-s − 0.455·17-s + 7.32·19-s − 5.23·21-s − 8.67·23-s − 0.242·25-s − 2.37·27-s − 3.71·29-s − 1.35·31-s + 4.34·35-s + 10.5·37-s − 2.62·39-s + 11.7·41-s + 1.27·43-s + 8.51·45-s + 4.96·47-s − 3.03·49-s + 1.19·51-s + 6.40·53-s − 19.2·57-s + 9.67·59-s − 5.68·61-s + ⋯ |
L(s) = 1 | − 1.51·3-s + 0.975·5-s + 0.752·7-s + 1.30·9-s + 0.277·13-s − 1.47·15-s − 0.110·17-s + 1.67·19-s − 1.14·21-s − 1.80·23-s − 0.0484·25-s − 0.457·27-s − 0.688·29-s − 0.242·31-s + 0.734·35-s + 1.73·37-s − 0.420·39-s + 1.83·41-s + 0.195·43-s + 1.26·45-s + 0.724·47-s − 0.433·49-s + 0.167·51-s + 0.879·53-s − 2.54·57-s + 1.25·59-s − 0.727·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.592214086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.592214086\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 - 1.99T + 7T^{2} \) |
| 17 | \( 1 + 0.455T + 17T^{2} \) |
| 19 | \( 1 - 7.32T + 19T^{2} \) |
| 23 | \( 1 + 8.67T + 23T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 31 | \( 1 + 1.35T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 - 4.96T + 47T^{2} \) |
| 53 | \( 1 - 6.40T + 53T^{2} \) |
| 59 | \( 1 - 9.67T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 + 1.70T + 67T^{2} \) |
| 71 | \( 1 + 7.76T + 71T^{2} \) |
| 73 | \( 1 - 7.82T + 73T^{2} \) |
| 79 | \( 1 + 5.87T + 79T^{2} \) |
| 83 | \( 1 + 9.81T + 83T^{2} \) |
| 89 | \( 1 - 2.62T + 89T^{2} \) |
| 97 | \( 1 - 2.67T + 97T^{2} \) |
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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
−7.75812161585712252182969517825, −7.36241328817233350199464436746, −6.25982008896065642807564632115, −5.81100615424652359815556739225, −5.49433280172733905489329218998, −4.61167619905314783814936447054, −3.91155610377752626159546046080, −2.52990393016443164202157024051, −1.61164385318737122433859663625, −0.75336042863862052947234240821,
0.75336042863862052947234240821, 1.61164385318737122433859663625, 2.52990393016443164202157024051, 3.91155610377752626159546046080, 4.61167619905314783814936447054, 5.49433280172733905489329218998, 5.81100615424652359815556739225, 6.25982008896065642807564632115, 7.36241328817233350199464436746, 7.75812161585712252182969517825