L(s) = 1 | + (0.0689 + 1.41i)2-s + (0.728 − 1.57i)3-s + (−1.99 + 0.194i)4-s + (1.00 + 1.73i)5-s + (2.27 + 0.920i)6-s + (2.50 + 0.837i)7-s + (−0.412 − 2.79i)8-s + (−1.93 − 2.28i)9-s + (−2.37 + 1.53i)10-s + (3.33 + 1.92i)11-s + (−1.14 + 3.26i)12-s + 2.05i·13-s + (−1.00 + 3.60i)14-s + (3.45 − 0.310i)15-s + (3.92 − 0.775i)16-s + (−4.92 − 2.84i)17-s + ⋯ |
L(s) = 1 | + (0.0487 + 0.998i)2-s + (0.420 − 0.907i)3-s + (−0.995 + 0.0973i)4-s + (0.447 + 0.775i)5-s + (0.926 + 0.375i)6-s + (0.948 + 0.316i)7-s + (−0.145 − 0.989i)8-s + (−0.646 − 0.762i)9-s + (−0.752 + 0.484i)10-s + (1.00 + 0.580i)11-s + (−0.330 + 0.943i)12-s + 0.571i·13-s + (−0.269 + 0.962i)14-s + (0.891 − 0.0802i)15-s + (0.981 − 0.193i)16-s + (−1.19 − 0.689i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21883 + 0.587198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21883 + 0.587198i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0689 - 1.41i)T \) |
| 3 | \( 1 + (-0.728 + 1.57i)T \) |
| 7 | \( 1 + (-2.50 - 0.837i)T \) |
good | 5 | \( 1 + (-1.00 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.33 - 1.92i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.05iT - 13T^{2} \) |
| 17 | \( 1 + (4.92 + 2.84i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.232 + 0.403i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.711 + 1.23i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9.25T + 29T^{2} \) |
| 31 | \( 1 + (4.99 + 2.88i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.569 + 0.328i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.42iT - 41T^{2} \) |
| 43 | \( 1 - 6.77T + 43T^{2} \) |
| 47 | \( 1 + (1.79 + 3.10i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.50 - 2.60i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.72 - 3.30i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.50 - 4.91i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.78 - 6.55i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + (2.02 - 3.50i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.00488 - 0.00281i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.56iT - 83T^{2} \) |
| 89 | \( 1 + (8.14 - 4.69i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.17T + 97T^{2} \) |
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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
−13.26906879885149665419083379715, −12.13621982110016406598507203715, −11.09870775837198768916435107639, −9.379033878580322788815918580611, −8.781710458579367965737270817623, −7.41645216081794887181714616271, −6.84322998440855193410607557378, −5.76141948181943555037045259649, −4.15694190844097891925159393111, −2.10726045148231320893132985378,
1.74832448184525958769794280039, 3.63508761227651265514927402610, 4.62734867011210901583202360310, 5.65764858462865057605091029265, 8.060264074821226932104908294763, 8.922521596058705994782740144395, 9.551468649473672327538735755644, 10.91282588368287922760603233530, 11.24117816131923962033300431179, 12.71304216945701399201505337828