| L(s) = 1 | + 1.69i·5-s + (−2.13 − 1.56i)7-s − 0.794i·11-s + 1.87i·13-s − 4.34i·17-s − 2.39·19-s + 3.62i·23-s + 2.12·25-s + 4.41·29-s − 1.87·31-s + (2.64 − 3.62i)35-s + 3.12·37-s − 4.34i·41-s + 0.876i·43-s − 12.0·47-s + ⋯ |
| L(s) = 1 | + 0.758i·5-s + (−0.807 − 0.590i)7-s − 0.239i·11-s + 0.519i·13-s − 1.05i·17-s − 0.550·19-s + 0.755i·23-s + 0.424·25-s + 0.820·29-s − 0.336·31-s + (0.447 − 0.612i)35-s + 0.513·37-s − 0.678i·41-s + 0.133i·43-s − 1.76·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.407818982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.407818982\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.13 + 1.56i)T \) |
| good | 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 11 | \( 1 + 0.794iT - 11T^{2} \) |
| 13 | \( 1 - 1.87iT - 13T^{2} \) |
| 17 | \( 1 + 4.34iT - 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 - 3.62iT - 23T^{2} \) |
| 29 | \( 1 - 4.41T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 4.34iT - 41T^{2} \) |
| 43 | \( 1 - 0.876iT - 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 0.876iT - 67T^{2} \) |
| 71 | \( 1 - 5.21iT - 71T^{2} \) |
| 73 | \( 1 - 3.74iT - 73T^{2} \) |
| 79 | \( 1 - 3.12iT - 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 7.73iT - 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
| 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
−8.625445587455581645381513728350, −7.62592041320020409686990353115, −6.96362528066001567284604032479, −6.58081845481495119144599592007, −5.69774182545088264476649990204, −4.72940267672583435843878230368, −3.83617511604670055479129807338, −3.11585019919862345844260349168, −2.31608636931077988322846635944, −0.863554347447107785135686898496,
0.52443554943272178688957741856, 1.82562071345257621897806965721, 2.81242007374276657559910190568, 3.70806436901921632197957565951, 4.63868131737207620907573646369, 5.30739599041519573069021857471, 6.24854113023385535890895998151, 6.61260735366744227901598723912, 7.77499443677625552579314052779, 8.497662522641446174462778083914
