L(s) = 1 | + (−0.5 + 0.866i)3-s + (−2.13 − 3.70i)5-s + (−1.5 + 2.17i)7-s + (−0.499 − 0.866i)9-s + (−2.13 + 3.70i)11-s + 1.27·13-s + 4.27·15-s + (2 − 3.46i)17-s + (0.637 + 1.10i)19-s + (−1.13 − 2.38i)21-s + (−2 − 3.46i)23-s + (−6.63 + 11.4i)25-s + 0.999·27-s + 2.27·29-s + (0.5 − 0.866i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.955 − 1.65i)5-s + (−0.566 + 0.823i)7-s + (−0.166 − 0.288i)9-s + (−0.644 + 1.11i)11-s + 0.353·13-s + 1.10·15-s + (0.485 − 0.840i)17-s + (0.146 + 0.253i)19-s + (−0.248 − 0.521i)21-s + (−0.417 − 0.722i)23-s + (−1.32 + 2.29i)25-s + 0.192·27-s + 0.422·29-s + (0.0898 − 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9537413007\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9537413007\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.5 - 2.17i)T \) |
good | 5 | \( 1 + (2.13 + 3.70i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.13 - 3.70i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.637 - 1.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.27T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.63 - 4.56i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.862 + 1.49i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.13 + 5.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.63 + 6.30i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (1.63 - 2.83i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.77 - 3.07i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.274T + 83T^{2} \) |
| 89 | \( 1 + (2.27 + 3.94i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.492108853799126749083694131222, −8.987635768511899244512328341227, −8.114649950514693523560341084469, −7.46405162869482342586618607414, −6.13134634249591319064980477679, −5.24725512063074691394726526676, −4.65657431717327280099050646353, −3.85233798314375505081059322936, −2.52105586965396797450994190197, −0.805934704748375686104086349315,
0.62176364172155249081991722309, 2.57536464287427881807438246442, 3.44048312551222311982981539421, 4.06830084517847150167052716824, 5.78695323371799730126126090951, 6.30192465828549041339079319791, 7.26560313272670563841206478103, 7.65158423490322427488285420124, 8.459743028988399890589922416841, 9.833814420090158632066879137439