L(s) = 1 | + (0.662 − 0.382i)2-s + (−0.207 + 0.358i)4-s + (0.5 + 0.866i)5-s + 1.08i·8-s + (0.662 + 0.382i)10-s + (0.207 + 0.358i)16-s + (−0.707 + 1.22i)17-s + (−0.662 + 0.382i)19-s − 0.414·20-s + (1.60 − 0.923i)23-s + (−0.499 + 0.866i)25-s + (−1.60 − 0.923i)31-s + (−0.662 − 0.382i)32-s + 1.08i·34-s + (−0.292 + 0.507i)38-s + ⋯ |
L(s) = 1 | + (0.662 − 0.382i)2-s + (−0.207 + 0.358i)4-s + (0.5 + 0.866i)5-s + 1.08i·8-s + (0.662 + 0.382i)10-s + (0.207 + 0.358i)16-s + (−0.707 + 1.22i)17-s + (−0.662 + 0.382i)19-s − 0.414·20-s + (1.60 − 0.923i)23-s + (−0.499 + 0.866i)25-s + (−1.60 − 0.923i)31-s + (−0.662 − 0.382i)32-s + 1.08i·34-s + (−0.292 + 0.507i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.562012107\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.562012107\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.258453402172154269014748215631, −8.715997674057188439449110294586, −7.80519615258655281464354347138, −6.96826048346597951192348073318, −6.14125102219744548151317317682, −5.39996727510964839364936765714, −4.33668097943153494392821034630, −3.67962435529752068404323242533, −2.69427760765912174344966729945, −1.95891701516378953690329134969,
0.900383713391440167506008717358, 2.20982604616486815209220465586, 3.56869403824301962049671979771, 4.51203250236470145404602738478, 5.26732733999436964391335681086, 5.56611854245317909992545460355, 6.85712433483714121612832396897, 7.11599259645637622783641212871, 8.601147752609035080077372992884, 9.067767156983061634757732641021