L(s) = 1 | + (−0.866 − 0.5i)2-s + (2.12 − 1.22i)3-s + (0.499 + 0.866i)4-s + (2.03 + 0.917i)5-s − 2.44·6-s − 0.999i·8-s + (1.49 − 2.59i)9-s + (−1.30 − 1.81i)10-s + (−2.44 − 4.24i)11-s + (2.12 + 1.22i)12-s − 0.449i·13-s + (5.44 − 0.550i)15-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (−2.59 + 1.49i)18-s + (3.22 − 5.58i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (1.22 − 0.707i)3-s + (0.249 + 0.433i)4-s + (0.911 + 0.410i)5-s − 0.999·6-s − 0.353i·8-s + (0.499 − 0.866i)9-s + (−0.413 − 0.573i)10-s + (−0.738 − 1.27i)11-s + (0.612 + 0.353i)12-s − 0.124i·13-s + (1.40 − 0.142i)15-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (−0.612 + 0.353i)18-s + (0.739 − 1.28i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52896 - 0.909526i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52896 - 0.909526i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.03 - 0.917i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-2.12 + 1.22i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.449iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.22 + 5.58i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.97 - 3.44i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.89T + 29T^{2} \) |
| 31 | \( 1 + (-0.449 - 0.778i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.73 - i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8.89iT - 43T^{2} \) |
| 47 | \( 1 + (0.778 + 0.449i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.953 - 0.550i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.22 + 5.58i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.22 - 7.31i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.92 + 4i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + (5.97 - 3.44i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.44 - 2.51i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.79iT - 97T^{2} \) |
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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
−10.78170801739162233225573740630, −9.708675074994379951395680056938, −8.946426581993900730602519252781, −8.309499391656264920586927383405, −7.34608061997004334047224912875, −6.52989441562695091645681924788, −5.21675363444942054918999153648, −3.10577359659057751806916458392, −2.77169755052971248251922067858, −1.35035387250007285635825959452,
1.83849057913812607088633996701, 2.91053091612169817590669046529, 4.48652234488148968773467436667, 5.36401104371708918610744544180, 6.72684665324158533882482196208, 7.74606234224310453030861373643, 8.651002830654775483350724956135, 9.264697189538180376231105371721, 10.07849363045936991357829888921, 10.40867480995570834314682252021