L(s) = 1 | + i·2-s + (0.939 + 1.45i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−1.45 + 0.939i)6-s + (2.57 − 0.617i)7-s − i·8-s + (−1.23 + 2.73i)9-s + (0.866 − 0.5i)10-s + (2.66 + 1.53i)11-s + (−0.939 − 1.45i)12-s + (0.449 + 0.259i)13-s + (0.617 + 2.57i)14-s + (0.790 − 1.54i)15-s + 16-s + (2.44 + 4.23i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.542 + 0.840i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.594 + 0.383i)6-s + (0.972 − 0.233i)7-s − 0.353i·8-s + (−0.411 + 0.911i)9-s + (0.273 − 0.158i)10-s + (0.803 + 0.463i)11-s + (−0.271 − 0.420i)12-s + (0.124 + 0.0720i)13-s + (0.164 + 0.687i)14-s + (0.204 − 0.397i)15-s + 0.250·16-s + (0.592 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00386 + 1.48640i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00386 + 1.48640i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.939 - 1.45i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.57 + 0.617i)T \) |
good | 11 | \( 1 + (-2.66 - 1.53i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.449 - 0.259i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.44 - 4.23i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.713 - 0.412i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.74 - 4.47i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.55 + 3.20i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 0.905iT - 31T^{2} \) |
| 37 | \( 1 + (-2.53 + 4.39i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.18 - 3.78i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.84 + 3.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + (4.69 - 2.71i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 9.52iT - 61T^{2} \) |
| 67 | \( 1 - 7.90T + 67T^{2} \) |
| 71 | \( 1 + 10.6iT - 71T^{2} \) |
| 73 | \( 1 + (-5.78 + 3.33i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + (-3.32 - 5.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.771 - 1.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.93 - 1.11i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.65445232542141532269774118591, −9.807422183784231193061598087339, −9.060139377108947374717869826999, −8.008928995603407925121517961068, −7.83249142308808245922939389943, −6.29909173661194314883293071162, −5.25497827644715957628073680512, −4.33054240824941016156749345528, −3.69576247729004685923499466798, −1.73220595739289301723379788640,
1.06031726173633999783707258745, 2.32487008231116775647269717130, 3.33714364317872848544107086665, 4.49962432632472348317011247356, 5.83981453036712675259665886752, 6.86680029462126689606813859350, 7.941243462285531787808936561362, 8.477842076972441294427316462244, 9.402009028061277440477847372331, 10.41238835635327793650592439858