| L(s) = 1 | + 2.90·3-s + 2·5-s − 0.779·7-s + 5.46·9-s + 5.03·11-s + 5.81·15-s − 6.46·17-s + 0.779·19-s − 2.26·21-s + 7.16·23-s − 25-s + 7.16·27-s + 3·29-s + 1.55·31-s + 14.6·33-s − 1.55·35-s + 4.26·37-s − 3.19·41-s − 8.72·43-s + 10.9·45-s + 7.37·47-s − 6.39·49-s − 18.8·51-s + 9.46·53-s + 10.0·55-s + 2.26·57-s − 12.4·59-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 0.894·5-s − 0.294·7-s + 1.82·9-s + 1.51·11-s + 1.50·15-s − 1.56·17-s + 0.178·19-s − 0.494·21-s + 1.49·23-s − 0.200·25-s + 1.37·27-s + 0.557·29-s + 0.280·31-s + 2.55·33-s − 0.263·35-s + 0.701·37-s − 0.499·41-s − 1.33·43-s + 1.62·45-s + 1.07·47-s − 0.913·49-s − 2.63·51-s + 1.29·53-s + 1.35·55-s + 0.300·57-s − 1.61·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.884169624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.884169624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 0.779T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 - 0.779T + 19T^{2} \) |
| 23 | \( 1 - 7.16T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 - 4.26T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 - 0.779T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 9.19T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.460695621636765761500106931762, −7.50262596352735598612383420848, −6.61205385110675913215127124042, −6.47256619802580000826229932857, −5.12677623494920046051003888956, −4.26664682453302570013059360108, −3.58056082701344505131959704587, −2.74533127673538015364014662278, −2.05611678814104365367289862196, −1.21248115921947163502407719856,
1.21248115921947163502407719856, 2.05611678814104365367289862196, 2.74533127673538015364014662278, 3.58056082701344505131959704587, 4.26664682453302570013059360108, 5.12677623494920046051003888956, 6.47256619802580000826229932857, 6.61205385110675913215127124042, 7.50262596352735598612383420848, 8.460695621636765761500106931762
