L(s) = 1 | + (−1.36 − 0.366i)2-s − 1.73·3-s + (1.73 + i)4-s − i·5-s + (2.36 + 0.633i)6-s + (−1.73 + 2i)7-s + (−1.99 − 2i)8-s + (−0.366 + 1.36i)10-s + 3.73i·11-s + (−2.99 − 1.73i)12-s + 6.46i·13-s + (3.09 − 2.09i)14-s + 1.73i·15-s + (1.99 + 3.46i)16-s + 0.464i·17-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s − 1.00·3-s + (0.866 + 0.5i)4-s − 0.447i·5-s + (0.965 + 0.258i)6-s + (−0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (−0.115 + 0.431i)10-s + 1.12i·11-s + (−0.866 − 0.499i)12-s + 1.79i·13-s + (0.827 − 0.560i)14-s + 0.447i·15-s + (0.499 + 0.866i)16-s + 0.112i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.161028 + 0.226198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.161028 + 0.226198i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 - 3.73iT - 11T^{2} \) |
| 13 | \( 1 - 6.46iT - 13T^{2} \) |
| 17 | \( 1 - 0.464iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 5.46iT - 23T^{2} \) |
| 29 | \( 1 + 5.92T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.53T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2.53iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 0.535iT - 71T^{2} \) |
| 73 | \( 1 + 0.928iT - 73T^{2} \) |
| 79 | \( 1 + 2.66iT - 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 - 9.46iT - 89T^{2} \) |
| 97 | \( 1 - 7.39iT - 97T^{2} \) |
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
−12.87045075431939220042488205749, −12.16799183960425978619245209917, −11.51440605557865281055700649435, −10.37026528910316763020881505267, −9.338165158708447121824484239823, −8.546921278297292523764169686316, −6.85869461782382583311609725357, −6.16062574620852869764995504444, −4.45755397677893659554666932901, −2.16481737214176128442862321924,
0.40432531224256162463675502881, 3.19256026462079209107629621232, 5.59676519147533025958951391265, 6.28296944808736258925591173451, 7.47112770983860742056380528090, 8.588311220882768324063685510937, 10.07624501647539433603349156003, 10.71805302863934986102624149781, 11.36904173487840485941988771164, 12.68054150275294888942475818803