| L(s) = 1 | + i·2-s + i·3-s − 4-s + (−2.20 − 0.344i)5-s − 6-s − i·8-s − 9-s + (0.344 − 2.20i)10-s + 6.10·11-s − i·12-s + 1.68i·13-s + (0.344 − 2.20i)15-s + 16-s + 6.83i·17-s − i·18-s + 1.68·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.988 − 0.153i)5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + (0.108 − 0.698i)10-s + 1.84·11-s − 0.288i·12-s + 0.468i·13-s + (0.0888 − 0.570i)15-s + 0.250·16-s + 1.65i·17-s − 0.235i·18-s + 0.387·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.038419928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038419928\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 + (2.20 + 0.344i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 - 1.68iT - 13T^{2} \) |
| 17 | \( 1 - 6.83iT - 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 23 | \( 1 + 2.31iT - 23T^{2} \) |
| 29 | \( 1 + 8.41T + 29T^{2} \) |
| 31 | \( 1 - 0.688T + 31T^{2} \) |
| 37 | \( 1 - 2.31iT - 37T^{2} \) |
| 41 | \( 1 + 5.14T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 3.06iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 + 12.2iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 - 13.5iT - 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 4.95iT - 97T^{2} \) |
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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
−9.563929150194169276087885654549, −9.007778876231027975899205171614, −8.341739229237694568666624309263, −7.50384016622520004299305351783, −6.58395968763843315040946250792, −5.94935990451832790421619689685, −4.69384067947268735732493724970, −4.02997512767432160506861697154, −3.46436355281777437526993226962, −1.42832564387342996842220239744,
0.46000313344161025565679255687, 1.65488935182073369090914466814, 3.07252777023641013411064095268, 3.72577420226629408537112399697, 4.72639632667525787010073614629, 5.79175314596819935723483660290, 7.00081086108741092952394331776, 7.37362595753765135939943217000, 8.440008645676218824266584686335, 9.164804909154349519143514410936
