L(s) = 1 | + (−0.390 + 1.68i)3-s + (1.54 − 0.894i)5-s + (−2.63 + 0.230i)7-s + (−2.69 − 1.31i)9-s + (−0.501 + 0.868i)11-s − 2.47·13-s + (0.904 + 2.96i)15-s + (−3.32 + 5.76i)17-s + (1.85 + 3.22i)19-s + (0.641 − 4.53i)21-s + (−6.85 + 3.95i)23-s + (−0.900 + 1.55i)25-s + (3.27 − 4.03i)27-s + 0.748·29-s + (−2.87 − 1.65i)31-s + ⋯ |
L(s) = 1 | + (−0.225 + 0.974i)3-s + (0.692 − 0.399i)5-s + (−0.996 + 0.0869i)7-s + (−0.898 − 0.439i)9-s + (−0.151 + 0.261i)11-s − 0.685·13-s + (0.233 + 0.765i)15-s + (−0.807 + 1.39i)17-s + (0.426 + 0.738i)19-s + (0.139 − 0.990i)21-s + (−1.42 + 0.824i)23-s + (−0.180 + 0.311i)25-s + (0.630 − 0.776i)27-s + 0.138·29-s + (−0.515 − 0.297i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0882405 + 0.632862i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0882405 + 0.632862i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.390 - 1.68i)T \) |
| 7 | \( 1 + (2.63 - 0.230i)T \) |
good | 5 | \( 1 + (-1.54 + 0.894i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.501 - 0.868i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 + (3.32 - 5.76i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.85 - 3.22i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.85 - 3.95i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.748T + 29T^{2} \) |
| 31 | \( 1 + (2.87 + 1.65i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.22 + 1.86i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 + 9.19iT - 43T^{2} \) |
| 47 | \( 1 + (1.19 + 2.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.33 - 10.9i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.34 - 4.24i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.02 - 3.50i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.89 + 3.98i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.46iT - 71T^{2} \) |
| 73 | \( 1 + (5.68 + 3.28i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.53 + 4.39i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 + (7.39 + 12.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.75iT - 97T^{2} \) |
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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
−10.55944187865729257934139288320, −9.978498943153193732857823467980, −9.415381496178092866235253862706, −8.606179193450600933427133088581, −7.38039013905329473697948207385, −5.96800071033309066465849490816, −5.73735617118488503613097754766, −4.38244463791618323725326562051, −3.52218515018123469629803894200, −2.08766025947324374730926437486,
0.31881077968978785945884451212, 2.26408622321145395629134353529, 2.97084767097954768636089154328, 4.73674869506152282318774748482, 5.88894149186487743440338236787, 6.58810353374315714270750022206, 7.21588376637944132812581444462, 8.274303447617980189614982707769, 9.414039502491089794331654485809, 9.987652461778234289165643496121