L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + (−0.5 − 0.866i)7-s + 0.999·8-s − 3·10-s + (3 + 5.19i)11-s + (0.5 − 0.866i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 3·17-s + 2·19-s + (1.50 + 2.59i)20-s + (3 − 5.19i)22-s + (3 − 5.19i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.188 − 0.327i)7-s + 0.353·8-s − 0.948·10-s + (0.904 + 1.56i)11-s + (0.138 − 0.240i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.727·17-s + 0.458·19-s + (0.335 + 0.580i)20-s + (0.639 − 1.10i)22-s + (0.625 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.607722610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.607722610\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.793009425611540517045828265720, −8.996558239787216145534504043210, −8.259221759748213239029838352112, −7.21732197553549228913214154370, −6.34165974762890287419742700325, −5.00854605935149852447247633123, −4.53479084820914783957455729668, −3.27536199925732988859333826106, −1.87430426293817310268732094758, −0.997023425202170479832554010303,
1.25317574174354436405884938233, 2.85175169230445470170579818927, 3.62751817610428634708668978409, 5.23040478913913337874608773361, 6.06612286073431944210998771369, 6.52971027584129180400570239012, 7.38635002088883832413786120331, 8.456095146412158674561356909972, 9.102048515020655962584395839390, 9.959412636747884270068846904767