L(s) = 1 | − i·3-s − 2.23i·5-s + 1.23i·7-s − 9-s − 2·11-s − 5.23i·13-s − 2.23·15-s + i·17-s − 6.47·19-s + 1.23·21-s + 4i·23-s − 5.00·25-s + i·27-s + 4·29-s − 2.47·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.999i·5-s + 0.467i·7-s − 0.333·9-s − 0.603·11-s − 1.45i·13-s − 0.577·15-s + 0.242i·17-s − 1.48·19-s + 0.269·21-s + 0.834i·23-s − 1.00·25-s + 0.192i·27-s + 0.742·29-s − 0.444·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2819479959\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2819479959\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + 2.23iT \) |
| 17 | \( 1 - iT \) |
good | 7 | \( 1 - 1.23iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 5.23iT - 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 7.70T + 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 1.70iT - 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.76iT - 73T^{2} \) |
| 79 | \( 1 - 0.944T + 79T^{2} \) |
| 83 | \( 1 - 10.4iT - 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 1.70iT - 97T^{2} \) |
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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
−8.356268082747924537089856823767, −8.154217623579506587427544614520, −7.27401757610306078317800045097, −6.28968987205691368975934624584, −5.59072527763094925640632060388, −5.09300375656129953604384451321, −4.10610048722845622415751108265, −3.03120219789291241890020678953, −2.15056542184496250130093119113, −1.10301304985688485936536160564,
0.082846234740168725565072584165, 1.98247643508177530252271915590, 2.67782010055908602708073282445, 3.79966083414469113715176292498, 4.27089181925853376177795912031, 5.17843912155961614641938952091, 6.19242628248492608153194271865, 6.81071349077508861342855292016, 7.32066156544698011912306538269, 8.434058776989840827254572558587