| L(s) = 1 | + 1.13i·3-s − 2.82i·5-s − 7-s + 1.70·9-s − 3.36i·11-s + i·13-s + 3.21·15-s − 2.33·17-s − 0.0742i·19-s − 1.13i·21-s + 4.61·23-s − 3.00·25-s + 5.35i·27-s − 1.06i·29-s + 6.88·31-s + ⋯ |
| L(s) = 1 | + 0.657i·3-s − 1.26i·5-s − 0.377·7-s + 0.568·9-s − 1.01i·11-s + 0.277i·13-s + 0.831·15-s − 0.566·17-s − 0.0170i·19-s − 0.248i·21-s + 0.962·23-s − 0.600·25-s + 1.03i·27-s − 0.197i·29-s + 1.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0840 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0840 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.518552322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.518552322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 - 1.13iT - 3T^{2} \) |
| 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 3.36iT - 11T^{2} \) |
| 17 | \( 1 + 2.33T + 17T^{2} \) |
| 19 | \( 1 + 0.0742iT - 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 + 1.06iT - 29T^{2} \) |
| 31 | \( 1 - 6.88T + 31T^{2} \) |
| 37 | \( 1 - 0.730iT - 37T^{2} \) |
| 41 | \( 1 + 5.51T + 41T^{2} \) |
| 43 | \( 1 + 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 + 4.63iT - 53T^{2} \) |
| 59 | \( 1 + 5.28iT - 59T^{2} \) |
| 61 | \( 1 - 1.47iT - 61T^{2} \) |
| 67 | \( 1 + 7.93iT - 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 0.246T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 9.61iT - 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.762895719628263367225554746293, −8.095383051164051948197418582433, −6.99178357100605868446381037767, −6.27432226051270112222757638402, −5.21653695129312386064666402658, −4.76128161209124171721337705288, −3.92651230306006485840488971554, −3.08509285686548973766131018793, −1.64859609285951826776377522095, −0.51166339201227551637211155102,
1.29496985614150606983293926452, 2.42487651758068140234725625176, 3.08039484425802852392910576933, 4.20375339418781344192342512744, 5.03328875347801244002378551340, 6.41266873867046591222415966814, 6.59552096360941464311786894704, 7.33553436155730295196267736485, 7.88724421105878861196441077923, 8.952705612887288241161326485220
