| L(s) = 1 | + i·2-s − 3-s − 4-s + (−2 + i)5-s − i·6-s − 2·7-s − i·8-s − 2·9-s + (−1 − 2i)10-s + 12-s + i·13-s − 2i·14-s + (2 − i)15-s + 16-s + (−1 + 4i)17-s − 2i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + (−0.894 + 0.447i)5-s − 0.408i·6-s − 0.755·7-s − 0.353i·8-s − 0.666·9-s + (−0.316 − 0.632i)10-s + 0.288·12-s + 0.277i·13-s − 0.534i·14-s + (0.516 − 0.258i)15-s + 0.250·16-s + (−0.242 + 0.970i)17-s − 0.471i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0315332 - 0.287260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0315332 - 0.287260i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 17 | \( 1 + (1 - 4i)T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 + 5iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 7iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 + 5iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 16iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−13.19512286103300492115631524185, −12.39050079814818103875850422595, −11.25689181617923631173042783422, −10.48558930738402783262927176327, −9.027226062417982679320643995668, −8.102116035085813470266632876031, −6.79457098757508507053371387289, −6.15832968561240465992565053715, −4.67218221910896069750003341173, −3.29356522318068060642161603631,
0.27902629164556144119710329257, 2.92968787802752899023078845282, 4.31829051465682132741427742833, 5.53796867274967089762237965035, 6.93565938255421960397609114452, 8.365504559433248474709823801345, 9.225353027840115778268430985145, 10.52435747411357564780555327956, 11.35065363829323746908868029367, 12.15195208048770882825759031227
