| L(s) = 1 | + 2-s + 0.255·3-s + 4-s − 5-s + 0.255·6-s + 8-s − 2.93·9-s − 10-s + 2.61·11-s + 0.255·12-s − 5.03·13-s − 0.255·15-s + 16-s − 17-s − 2.93·18-s + 2.49·19-s − 20-s + 2.61·22-s − 0.0298·23-s + 0.255·24-s + 25-s − 5.03·26-s − 1.51·27-s + 4.55·29-s − 0.255·30-s − 3.07·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.147·3-s + 0.5·4-s − 0.447·5-s + 0.104·6-s + 0.353·8-s − 0.978·9-s − 0.316·10-s + 0.787·11-s + 0.0736·12-s − 1.39·13-s − 0.0658·15-s + 0.250·16-s − 0.242·17-s − 0.691·18-s + 0.571·19-s − 0.223·20-s + 0.556·22-s − 0.00621·23-s + 0.0520·24-s + 0.200·25-s − 0.988·26-s − 0.291·27-s + 0.846·29-s − 0.0465·30-s − 0.551·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.616517602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.616517602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.255T + 3T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 + 0.0298T + 23T^{2} \) |
| 29 | \( 1 - 4.55T + 29T^{2} \) |
| 31 | \( 1 + 3.07T + 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 + 5.09T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 - 7.02T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.95T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 1.30T + 89T^{2} \) |
| 97 | \( 1 + 3.38T + 97T^{2} \) |
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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
−7.76826354225569279818754878505, −6.94362171141601234631089248783, −6.53374778167778761775878382571, −5.51225802340929789577480545901, −5.07488400369971938930458106160, −4.23391516316231390112468514919, −3.52780451623094584633496533181, −2.77107102612845320320847040185, −2.06432271582936837753096502327, −0.68439813757926907152184894986,
0.68439813757926907152184894986, 2.06432271582936837753096502327, 2.77107102612845320320847040185, 3.52780451623094584633496533181, 4.23391516316231390112468514919, 5.07488400369971938930458106160, 5.51225802340929789577480545901, 6.53374778167778761775878382571, 6.94362171141601234631089248783, 7.76826354225569279818754878505
