L(s) = 1 | + i·2-s + i·3-s − 4-s + (−0.0743 − 2.23i)5-s − 6-s − i·8-s − 9-s + (2.23 − 0.0743i)10-s − 3.05·11-s − i·12-s + 1.64i·13-s + (2.23 − 0.0743i)15-s + 16-s + 2.90i·17-s − i·18-s + 2.21·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.0332 − 0.999i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.706 − 0.0234i)10-s − 0.921·11-s − 0.288i·12-s + 0.455i·13-s + (0.577 − 0.0191i)15-s + 0.250·16-s + 0.705i·17-s − 0.235i·18-s + 0.507·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4454737615\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4454737615\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.0743 + 2.23i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 - 1.64iT - 13T^{2} \) |
| 17 | \( 1 - 2.90iT - 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 + 1.78iT - 23T^{2} \) |
| 29 | \( 1 + 5.58T + 29T^{2} \) |
| 31 | \( 1 + 0.944T + 31T^{2} \) |
| 37 | \( 1 + 10.7iT - 37T^{2} \) |
| 41 | \( 1 + 6.67T + 41T^{2} \) |
| 43 | \( 1 + 10.5iT - 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 + 5.78iT - 53T^{2} \) |
| 59 | \( 1 + 5.97T + 59T^{2} \) |
| 61 | \( 1 + 0.445T + 61T^{2} \) |
| 67 | \( 1 - 1.26iT - 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 7.91iT - 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 0.874iT - 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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
−9.077228364775095044100566393528, −8.608631046858185406568015198973, −7.78824978309022008793788859503, −6.96440448219150267072046959528, −5.65851455464192535517075244889, −5.35303315241105025425312202972, −4.34702461362815510269757591836, −3.58930639220941951172968560565, −1.96654460593474430514405208819, −0.17312015143359865980274039355,
1.52595137906507846856171228040, 2.82929012139657954657998159400, 3.16955703480934240845740312216, 4.62355396925195071472126932384, 5.59858366265076175201060576528, 6.44540003506836115284821151686, 7.58520658465862530517981925834, 7.80565500952950275915548105406, 9.041903828236924368170442926536, 9.894100796212435314572512292736