| L(s) = 1 | + i·2-s + 4-s − 4i·7-s + 3i·8-s + 3·9-s − 2i·13-s + 4·14-s − 16-s − i·17-s + 3i·18-s + 4·19-s + 4i·23-s + 2·26-s − 4i·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.5·4-s − 1.51i·7-s + 1.06i·8-s + 9-s − 0.554i·13-s + 1.06·14-s − 0.250·16-s − 0.242i·17-s + 0.707i·18-s + 0.917·19-s + 0.834i·23-s + 0.392·26-s − 0.755i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69010 + 0.398979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69010 + 0.398979i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−11.12545955184019786026691703065, −10.34419994153612588694837079470, −9.580103581843853268273104393610, −8.054755129932527360045391104644, −7.37323907538424822017392644714, −6.88283093269140417083303385877, −5.64558969832405997965298070006, −4.50562702493167937967098019859, −3.27421846633199865520241686865, −1.39699613843806692195408059085,
1.66415178220154260649314063962, 2.67920169036367087688039689822, 3.96408590383967063163987482962, 5.34179206358898239150626680022, 6.43689548753242383069829388258, 7.30700044211496192342148109860, 8.571556531784596602191283720981, 9.526145800625271601965973157201, 10.20379975024811324405866717620, 11.30343291001595261223671925186
