L(s) = 1 | + i·2-s + (1.20 − 1.24i)3-s − 4-s + (0.866 + 0.5i)5-s + (1.24 + 1.20i)6-s + (2.16 + 3.75i)7-s − i·8-s + (−0.110 − 2.99i)9-s + (−0.5 + 0.866i)10-s + (2.88 − 5.00i)11-s + (−1.20 + 1.24i)12-s + (0.829 + 0.478i)13-s + (−3.75 + 2.16i)14-s + (1.66 − 0.478i)15-s + 16-s + (−0.169 − 0.293i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.694 − 0.719i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (0.509 + 0.490i)6-s + (0.819 + 1.42i)7-s − 0.353i·8-s + (−0.0367 − 0.999i)9-s + (−0.158 + 0.273i)10-s + (0.871 − 1.50i)11-s + (−0.347 + 0.359i)12-s + (0.230 + 0.132i)13-s + (−1.00 + 0.579i)14-s + (0.429 − 0.123i)15-s + 0.250·16-s + (−0.0410 − 0.0710i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.23647 + 0.571283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.23647 + 0.571283i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.20 + 1.24i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (-4.75 - 2.90i)T \) |
good | 7 | \( 1 + (-2.16 - 3.75i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.88 + 5.00i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.829 - 0.478i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.169 + 0.293i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.915 - 1.58i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.13T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 37 | \( 1 + (5.82 - 3.36i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.47 - 0.849i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.67 + 2.12i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.77iT - 47T^{2} \) |
| 53 | \( 1 + (4.01 - 6.95i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.461 + 0.266i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 12.6iT - 61T^{2} \) |
| 67 | \( 1 + (-6.77 + 11.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.72 - 3.30i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.51 - 3.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.30 - 1.33i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.17 + 7.22i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 0.143T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−9.755636718152734066004961083623, −8.910603736202704354370022108065, −8.409652299825255131053906691225, −7.88050182086304933358033070940, −6.36007652769593753718792728364, −6.24639471202130333036422397215, −5.14760310507461891996289059811, −3.70430150216372752034684067234, −2.61144102372590300138934423043, −1.36736879045975593200729343295,
1.35583017605986221532301021582, 2.37598153260361050744312213127, 3.95361355135173220675350586381, 4.27977408704746612859229714651, 5.15934842418503429254402036565, 6.74465163215934560913533571355, 7.72553326418024871747836818137, 8.445622283743137704370940794705, 9.459775771136232866925054363595, 10.07323869538125857993219041804