L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.340 + 0.589i)3-s + (−0.499 + 0.866i)4-s + (1.37 + 1.76i)5-s + 0.681·6-s + (2.52 + 0.782i)7-s + 0.999·8-s + (1.26 + 2.19i)9-s + (0.839 − 2.07i)10-s + (−3.24 + 0.708i)11-s + (−0.340 − 0.589i)12-s + 2.05i·13-s + (−0.585 − 2.58i)14-s + (−1.50 + 0.210i)15-s + (−0.5 − 0.866i)16-s + (2.89 + 1.67i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.196 + 0.340i)3-s + (−0.249 + 0.433i)4-s + (0.614 + 0.788i)5-s + 0.278·6-s + (0.955 + 0.295i)7-s + 0.353·8-s + (0.422 + 0.732i)9-s + (0.265 − 0.655i)10-s + (−0.976 + 0.213i)11-s + (−0.0982 − 0.170i)12-s + 0.568i·13-s + (−0.156 − 0.689i)14-s + (−0.389 + 0.0543i)15-s + (−0.125 − 0.216i)16-s + (0.701 + 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04359 + 0.735065i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04359 + 0.735065i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-1.37 - 1.76i)T \) |
| 7 | \( 1 + (-2.52 - 0.782i)T \) |
| 11 | \( 1 + (3.24 - 0.708i)T \) |
good | 3 | \( 1 + (0.340 - 0.589i)T + (-1.5 - 2.59i)T^{2} \) |
| 13 | \( 1 - 2.05iT - 13T^{2} \) |
| 17 | \( 1 + (-2.89 - 1.67i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.31 + 4.00i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.24 - 1.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.208iT - 29T^{2} \) |
| 31 | \( 1 + (7.69 + 4.44i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.14 + 1.81i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.23T + 41T^{2} \) |
| 43 | \( 1 - 6.55T + 43T^{2} \) |
| 47 | \( 1 + (-6.57 - 11.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.20 + 1.84i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.99 - 2.88i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.38 - 9.32i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.44 + 0.836i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.74T + 71T^{2} \) |
| 73 | \( 1 + (6.62 + 3.82i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.3 + 7.71i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.84iT - 83T^{2} \) |
| 89 | \( 1 + (-15.0 + 8.67i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
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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
−10.71370513269941108815894922998, −9.809835748389364324104486664227, −9.018617678605286132761729491301, −7.83892256418208063725503293577, −7.34488117340759175988926781626, −5.89629531669528955482903711115, −5.04468197619501047857059010562, −4.05425814160460174091159522714, −2.54778828097305169254206874699, −1.83361389184751005823847673509,
0.76238324479436689116786175250, 1.94931578860110714954180281340, 3.86773430868326598635621199380, 5.16033932307033525467492738575, 5.59453533091276323302875853263, 6.69371401383485089373981017340, 7.75018660939858962985317470997, 8.244485317003906116240744022467, 9.195544274034314258176342177483, 10.12794555843803270780626859359