| L(s) = 1 | + 0.610i·3-s + 0.780·5-s + (−2.18 − 1.49i)7-s + 2.62·9-s − 3.13·11-s − 13-s + 0.476i·15-s + 4.47i·17-s + 1.92i·19-s + (0.910 − 1.33i)21-s − 5.54i·23-s − 4.39·25-s + 3.43i·27-s + 2.44i·29-s + 5.76·31-s + ⋯ |
| L(s) = 1 | + 0.352i·3-s + 0.349·5-s + (−0.826 − 0.563i)7-s + 0.875·9-s − 0.945·11-s − 0.277·13-s + 0.123i·15-s + 1.08i·17-s + 0.442i·19-s + (0.198 − 0.291i)21-s − 1.15i·23-s − 0.878·25-s + 0.661i·27-s + 0.453i·29-s + 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3762200641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3762200641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.18 + 1.49i)T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.610iT - 3T^{2} \) |
| 5 | \( 1 - 0.780T + 5T^{2} \) |
| 11 | \( 1 + 3.13T + 11T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 1.92iT - 19T^{2} \) |
| 23 | \( 1 + 5.54iT - 23T^{2} \) |
| 29 | \( 1 - 2.44iT - 29T^{2} \) |
| 31 | \( 1 - 5.76T + 31T^{2} \) |
| 37 | \( 1 + 0.0497iT - 37T^{2} \) |
| 41 | \( 1 + 9.71iT - 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 7.60iT - 53T^{2} \) |
| 59 | \( 1 - 13.2iT - 59T^{2} \) |
| 61 | \( 1 + 8.35T + 61T^{2} \) |
| 67 | \( 1 + 0.937T + 67T^{2} \) |
| 71 | \( 1 + 13.0iT - 71T^{2} \) |
| 73 | \( 1 + 0.406iT - 73T^{2} \) |
| 79 | \( 1 - 4.03iT - 79T^{2} \) |
| 83 | \( 1 - 11.9iT - 83T^{2} \) |
| 89 | \( 1 - 15.7iT - 89T^{2} \) |
| 97 | \( 1 + 7.10iT - 97T^{2} \) |
| 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
−9.269240546263784296067246770637, −8.296050155621525179916846943562, −7.63379235906507805146196118007, −6.75298664134391561084120477154, −6.17042453226193942665526715901, −5.20412205482713315566148490944, −4.34819696644624558572357265453, −3.63642779096562740904504168042, −2.62561934176873696375450630814, −1.47502477862430871440778215067,
0.11287216286074271115376490943, 1.65460320972884907930327705992, 2.63448233708710126339260272837, 3.38352643464068149752535657417, 4.69545775122160197785186565757, 5.28556384804988131420951535069, 6.28035431256227998699107141189, 6.81508661865081853369315860413, 7.68704288168274107684418369451, 8.250270736901497870039753368596
