| L(s) = 1 | − 0.665i·5-s − i·7-s − 2.07·11-s − 5.55·13-s − 2.16i·17-s − 4.49i·19-s + 4.28·23-s + 4.55·25-s − 1.41i·29-s + 6.61i·31-s − 0.665·35-s + 5.43·37-s − 5.69i·41-s + 2.11i·43-s − 10.5·47-s + ⋯ |
| L(s) = 1 | − 0.297i·5-s − 0.377i·7-s − 0.627·11-s − 1.54·13-s − 0.524i·17-s − 1.03i·19-s + 0.892·23-s + 0.911·25-s − 0.262i·29-s + 1.18i·31-s − 0.112·35-s + 0.894·37-s − 0.889i·41-s + 0.322i·43-s − 1.53·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06798823916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06798823916\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 + 0.665iT - 5T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 5.55T + 13T^{2} \) |
| 17 | \( 1 + 2.16iT - 17T^{2} \) |
| 19 | \( 1 + 4.49iT - 19T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.61iT - 31T^{2} \) |
| 37 | \( 1 - 5.43T + 37T^{2} \) |
| 41 | \( 1 + 5.69iT - 41T^{2} \) |
| 43 | \( 1 - 2.11iT - 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 0.615T + 61T^{2} \) |
| 67 | \( 1 - 14.0iT - 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 0.824iT - 79T^{2} \) |
| 83 | \( 1 - 6.36T + 83T^{2} \) |
| 89 | \( 1 - 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 0.824T + 97T^{2} \) |
| 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
−8.872604953408134661196709086890, −7.88053172783892856455640768320, −7.24155622120168685525991579339, −6.78419297108960153951585830552, −5.62528530036187098080300311171, −4.78692656418839317553794671906, −4.56793232711051310392813215302, −3.04920680190547790969977675558, −2.60200613790308956010116932702, −1.19015302654336680262086316331,
0.01972896319906616089865538733, 1.66299146394683237871553114404, 2.64640438278198415028782047245, 3.29146752209230988894789142770, 4.52969185698805292843167467780, 5.07429278046845897150304361858, 5.96370931804924973030115093241, 6.65549157382613034555779863073, 7.58949103989693602000561800764, 7.952892309007474536074179865659
