| L(s) = 1 | + 2.23·5-s + i·7-s + 2·11-s − 4.47i·13-s + 2.47i·17-s − 2·19-s + 4i·23-s + 5.00·25-s + 0.472·29-s + 8.47·31-s + 2.23i·35-s − 6.47i·37-s + 12.4·41-s − 6.47i·43-s + 2.47i·47-s + ⋯ |
| L(s) = 1 | + 0.999·5-s + 0.377i·7-s + 0.603·11-s − 1.24i·13-s + 0.599i·17-s − 0.458·19-s + 0.834i·23-s + 1.00·25-s + 0.0876·29-s + 1.52·31-s + 0.377i·35-s − 1.06i·37-s + 1.94·41-s − 0.986i·43-s + 0.360i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.348447413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.348447413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 6.47iT - 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.47iT - 43T^{2} \) |
| 47 | \( 1 - 2.47iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 10.4iT - 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 + 16.4iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−9.065779613500727244565554067406, −8.189302660565129662404860382400, −7.43919359581107216300344638318, −6.32435016968409254855350217916, −5.92277524434864579312820268897, −5.14515283166051883873038889939, −4.11062186266416378142198170852, −3.02848833611702520020621631106, −2.16082444557884008439813359293, −1.01294888037099936433042195482,
1.04021977116270304535962671624, 2.09964353407427040330629333812, 3.02688131945124668336337135387, 4.37135113917201582958858077952, 4.74989833386195693232903285521, 6.10853163606682044385649771248, 6.44478025107291075797943366563, 7.22152827314317103354901563013, 8.268840060525522793311029857032, 9.059655210536636436954010239971
