| L(s) = 1 | + i·3-s + (2 − i)5-s − 9-s − 4·11-s + 6i·13-s + (1 + 2i)15-s + 2i·17-s + 6·19-s + 2i·23-s + (3 − 4i)25-s − i·27-s − 6·29-s + 2·31-s − 4i·33-s + 4i·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.894 − 0.447i)5-s − 0.333·9-s − 1.20·11-s + 1.66i·13-s + (0.258 + 0.516i)15-s + 0.485i·17-s + 1.37·19-s + 0.417i·23-s + (0.600 − 0.800i)25-s − 0.192i·27-s − 1.11·29-s + 0.359·31-s − 0.696i·33-s + 0.657i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.513279327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.513279327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
| 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
−9.193312081905547956992625821414, −8.389178905416906350349143839349, −7.53512869709180877756104293968, −6.64753181213099976954636435229, −5.75294304312406214492921052825, −5.16246495904395590380259888160, −4.45641711568603235735339652398, −3.41931128270595660758814048524, −2.37345130791970759533209260897, −1.42697032789106415059739198003,
0.45739985750176286797898739644, 1.79453872398502107467340572743, 2.83967050039489236148535193732, 3.26331334849043366426550215891, 5.00619070025271016761415638568, 5.47488415821973198647667497892, 6.10953607522615709092679695626, 7.17438565335602388142547460088, 7.62885576755961046890038129304, 8.354692515824905108254626759614
