L(s) = 1 | + (−0.707 + 1.58i)3-s − 1.41i·5-s − i·7-s + (−2.00 − 2.23i)9-s + 4.47·11-s − 7.16·13-s + (2.23 + 1.00i)15-s − 7.30i·17-s − 0.837i·19-s + (1.58 + 0.707i)21-s − 5.65·23-s + 2.99·25-s + (4.94 − 1.58i)27-s − 1.64i·29-s − 6.32i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.912i)3-s − 0.632i·5-s − 0.377i·7-s + (−0.666 − 0.745i)9-s + 1.34·11-s − 1.98·13-s + (0.577 + 0.258i)15-s − 1.77i·17-s − 0.192i·19-s + (0.345 + 0.154i)21-s − 1.17·23-s + 0.599·25-s + (0.952 − 0.304i)27-s − 0.305i·29-s − 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.784574 - 0.540179i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.784574 - 0.540179i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 1.58i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 7.16T + 13T^{2} \) |
| 17 | \( 1 + 7.30iT - 17T^{2} \) |
| 19 | \( 1 + 0.837iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 1.64iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 - 8.32iT - 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 1.18iT - 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 7.53T + 83T^{2} \) |
| 89 | \( 1 + 1.18iT - 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08903260616041100372782625664, −9.521644848619921505011598911496, −9.014574462546072411829455081747, −7.65379320743729376734457195517, −6.78706729051840360042628597273, −5.60158573770289924988041630305, −4.68774349523436148611786533459, −4.10242308405640643411985180278, −2.61179994072397058484919306686, −0.53611211424085824405630853413,
1.62982171533363401867850436269, 2.72714324522088687159536663149, 4.17876008620156716390738872404, 5.46890981987341229110392590609, 6.39757924291938086964494317383, 6.99135593157556360983296951832, 7.920073957311080121122214460560, 8.829312221788833833934588043961, 9.956966055858821857280571567796, 10.69600003646635421377497429262