L(s) = 1 | + (−0.938 − 1.05i)2-s + (−0.219 − 0.126i)3-s + (−0.239 + 1.98i)4-s + (−0.5 − 0.866i)5-s + (0.0718 + 0.351i)6-s + (0.978 + 2.45i)7-s + (2.32 − 1.60i)8-s + (−1.46 − 2.54i)9-s + (−0.447 + 1.34i)10-s + (1.81 − 3.14i)11-s + (0.304 − 0.405i)12-s + 5.36·13-s + (1.68 − 3.34i)14-s + 0.253i·15-s + (−3.88 − 0.951i)16-s + (−4.46 − 2.58i)17-s + ⋯ |
L(s) = 1 | + (−0.663 − 0.748i)2-s + (−0.126 − 0.0731i)3-s + (−0.119 + 0.992i)4-s + (−0.223 − 0.387i)5-s + (0.0293 + 0.143i)6-s + (0.369 + 0.929i)7-s + (0.822 − 0.569i)8-s + (−0.489 − 0.847i)9-s + (−0.141 + 0.424i)10-s + (0.547 − 0.947i)11-s + (0.0877 − 0.117i)12-s + 1.48·13-s + (0.449 − 0.893i)14-s + 0.0654i·15-s + (−0.971 − 0.237i)16-s + (−1.08 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0638 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0638 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.636543 - 0.597113i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.636543 - 0.597113i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.938 + 1.05i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.978 - 2.45i)T \) |
good | 3 | \( 1 + (0.219 + 0.126i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.81 + 3.14i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 + (4.46 + 2.58i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.49 + 3.17i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.231 + 0.133i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.99iT - 29T^{2} \) |
| 31 | \( 1 + (-2.72 + 4.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.48 - 4.32i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 5.33T + 43T^{2} \) |
| 47 | \( 1 + (-2.26 - 3.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.09 - 1.78i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.83 - 3.94i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.63 + 4.55i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.963 + 1.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 15.9iT - 71T^{2} \) |
| 73 | \( 1 + (-7.12 - 4.11i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.94 - 5.74i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.75iT - 83T^{2} \) |
| 89 | \( 1 + (-4.77 + 2.75i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7.98iT - 97T^{2} \) |
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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
−11.47448928858627397920297615487, −11.12188789323935986642281284499, −9.443371538638938916052035822841, −8.855374412889453490541688189343, −8.285424441145472428779990208183, −6.76061776064460250616295460273, −5.59620105184587432445121777658, −4.00498139098755459612521921211, −2.82787010853863755566966283362, −0.964523253517324158727950035630,
1.59613522216611778274127551136, 3.92005790049279766691342615719, 5.12908728533062653822869144734, 6.39050878818784027613569383935, 7.26743919727614206268664599464, 8.141216318251883676507963011743, 9.048814755912746603099392136791, 10.40082739695690897775898019380, 10.76616048072884045689355309527, 11.74254799266068381825224633014