| L(s) = 1 | − i·2-s − 4-s + (1 − 2i)5-s − i·7-s + i·8-s + (−2 − i)10-s + 6·11-s − 2i·13-s − 14-s + 16-s + 2i·17-s − 4·19-s + (−1 + 2i)20-s − 6i·22-s − 4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.447 − 0.894i)5-s − 0.377i·7-s + 0.353i·8-s + (−0.632 − 0.316i)10-s + 1.80·11-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s + 0.485i·17-s − 0.917·19-s + (−0.223 + 0.447i)20-s − 1.27i·22-s − 0.834i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.812169 - 1.31411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.812169 - 1.31411i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−10.39094697653692057817685681562, −9.322654556411110186300742997320, −8.930246933645171070402983337038, −7.922833410990401390875616583805, −6.57705290408452007663423881668, −5.69666066139960334794565233813, −4.42263011885262614600908114881, −3.80379075719330917268802144199, −2.11128245900363896438612591366, −0.928398710149656900554118775543,
1.78645075304316138957678195879, 3.35291355479380674543971384468, 4.40069617765707568060614982719, 5.71922739888139986180958741545, 6.53880093670106255874040911690, 7.01494896659302406705491571136, 8.224895312960045316519149718886, 9.323450849642180450363379040455, 9.598799220156375974774715569831, 10.93331767654709263824327326425
