| L(s) = 1 | − 2-s + 4-s − 0.968i·5-s − i·7-s − 8-s + 0.968i·10-s + (3.27 − 0.506i)11-s − 6.36i·13-s + i·14-s + 16-s − 7.67·17-s − 0.690i·19-s − 0.968i·20-s + (−3.27 + 0.506i)22-s + 3.04i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.433i·5-s − 0.377i·7-s − 0.353·8-s + 0.306i·10-s + (0.988 − 0.152i)11-s − 1.76i·13-s + 0.267i·14-s + 0.250·16-s − 1.86·17-s − 0.158i·19-s − 0.216i·20-s + (−0.698 + 0.107i)22-s + 0.634i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7531854435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7531854435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-3.27 + 0.506i)T \) |
| good | 5 | \( 1 + 0.968iT - 5T^{2} \) |
| 13 | \( 1 + 6.36iT - 13T^{2} \) |
| 17 | \( 1 + 7.67T + 17T^{2} \) |
| 19 | \( 1 + 0.690iT - 19T^{2} \) |
| 23 | \( 1 - 3.04iT - 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 + 4.16T + 37T^{2} \) |
| 41 | \( 1 - 0.349T + 41T^{2} \) |
| 43 | \( 1 - 0.794iT - 43T^{2} \) |
| 47 | \( 1 + 8.19iT - 47T^{2} \) |
| 53 | \( 1 - 7.24iT - 53T^{2} \) |
| 59 | \( 1 + 12.0iT - 59T^{2} \) |
| 61 | \( 1 - 8.29iT - 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 4.98iT - 71T^{2} \) |
| 73 | \( 1 + 9.97iT - 73T^{2} \) |
| 79 | \( 1 + 2.37iT - 79T^{2} \) |
| 83 | \( 1 + 4.79T + 83T^{2} \) |
| 89 | \( 1 + 7.23iT - 89T^{2} \) |
| 97 | \( 1 + 2.78T + 97T^{2} \) |
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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
−9.069930751891879124703484204220, −8.684218674980474435338046794618, −7.71752489321776702538909922588, −6.97480248918877395363903704552, −6.10112193963292676779475116635, −5.14026376058400374338931086581, −4.04576164494588269456667605550, −2.99276814936019867875283539035, −1.63760399211369815509140326675, −0.38139635885662654584372714432,
1.64826557643766270355034738685, 2.48453579007152069048382815395, 3.88798763341844693352860042767, 4.69831471544069787967840739308, 6.19027699425321522847881627972, 6.68926714671418995151119440830, 7.29580527140319453917256068241, 8.623705217800639255237451710532, 9.022031075664352759251690132770, 9.626412070206768450778598556637
