| L(s) = 1 | + (−1.35 − 0.411i)2-s + (−1.86 + 0.994i)3-s + (1.66 + 1.11i)4-s + (−0.634 − 0.773i)5-s + (2.92 − 0.579i)6-s + (4.23 − 2.82i)7-s + (−1.78 − 2.19i)8-s + (0.806 − 1.20i)9-s + (0.539 + 1.30i)10-s + (−2.83 + 0.861i)11-s + (−4.19 − 0.422i)12-s + (−0.987 − 0.810i)13-s + (−6.89 + 2.08i)14-s + (1.94 + 0.807i)15-s + (1.51 + 3.70i)16-s + (2.25 − 0.932i)17-s + ⋯ |
| L(s) = 1 | + (−0.956 − 0.291i)2-s + (−1.07 + 0.574i)3-s + (0.830 + 0.557i)4-s + (−0.283 − 0.345i)5-s + (1.19 − 0.236i)6-s + (1.60 − 1.06i)7-s + (−0.631 − 0.775i)8-s + (0.268 − 0.402i)9-s + (0.170 + 0.413i)10-s + (−0.855 + 0.259i)11-s + (−1.21 − 0.122i)12-s + (−0.273 − 0.224i)13-s + (−1.84 + 0.556i)14-s + (0.503 + 0.208i)15-s + (0.378 + 0.925i)16-s + (0.545 − 0.226i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.253749 - 0.363182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.253749 - 0.363182i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 + 0.411i)T \) |
| 5 | \( 1 + (0.634 + 0.773i)T \) |
| good | 3 | \( 1 + (1.86 - 0.994i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (-4.23 + 2.82i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (2.83 - 0.861i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (0.987 + 0.810i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-2.25 + 0.932i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.0107 + 0.109i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (2.85 - 0.568i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (2.35 - 7.77i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (5.29 + 5.29i)T + 31iT^{2} \) |
| 37 | \( 1 + (-10.2 - 1.01i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.74 + 8.77i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (9.78 + 5.23i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (0.994 + 2.40i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (3.07 + 10.1i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-4.74 + 3.89i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-1.87 - 3.51i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (5.55 + 10.3i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (1.44 + 2.15i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-0.437 - 0.292i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (2.88 - 6.97i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (3.39 - 0.334i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.30 - 1.05i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-4.29 - 4.29i)T + 97iT^{2} \) |
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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
−10.41378129909217170577683382279, −9.798823535970589694171369126740, −8.438049840602765791890584108120, −7.77726422407787532655644197615, −7.10870995994180674090100224913, −5.54104030106539209117023453617, −4.85367229026385954452570170715, −3.77038165251977418125773404819, −1.87179986970206643520349235525, −0.39230471373502785995033451148,
1.41846227269678947247583292900, 2.60283453293689951331294380359, 4.85044595078465388039521627689, 5.69503483942906963926378912353, 6.29403367202944525971950598561, 7.63988322632506180260108970899, 7.939476305190510107621651344388, 8.935425353327202428512627695037, 10.10003846727584487064093973122, 11.02079989751879679288398700819
