L(s) = 1 | + (1.92 + 3.32i)5-s + (2.55 + 0.693i)7-s + (−0.903 + 1.56i)11-s + (−0.692 + 1.19i)13-s + (0.833 + 1.44i)17-s + (0.0802 − 0.138i)19-s + (−1.60 − 2.77i)23-s + (−4.87 + 8.44i)25-s + (3.78 + 6.54i)29-s − 3.22·31-s + (2.59 + 9.82i)35-s + (1.58 − 2.74i)37-s + (−6.00 + 10.3i)41-s + (−3.45 − 5.98i)43-s + 11.4·47-s + ⋯ |
L(s) = 1 | + (0.858 + 1.48i)5-s + (0.965 + 0.261i)7-s + (−0.272 + 0.471i)11-s + (−0.192 + 0.332i)13-s + (0.202 + 0.350i)17-s + (0.0184 − 0.0318i)19-s + (−0.333 − 0.577i)23-s + (−0.975 + 1.68i)25-s + (0.701 + 1.21i)29-s − 0.578·31-s + (0.439 + 1.66i)35-s + (0.260 − 0.451i)37-s + (−0.937 + 1.62i)41-s + (−0.526 − 0.912i)43-s + 1.66·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.226512188\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.226512188\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.55 - 0.693i)T \) |
good | 5 | \( 1 + (-1.92 - 3.32i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.903 - 1.56i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.692 - 1.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.833 - 1.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0802 + 0.138i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.60 + 2.77i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.78 - 6.54i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 + (-1.58 + 2.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.00 - 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.45 + 5.98i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + (1.37 + 2.38i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 15.0T + 59T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 6.93T + 71T^{2} \) |
| 73 | \( 1 + (6.22 + 10.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + (1.45 + 2.51i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.04 - 8.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.18 - 7.25i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.945010143834233598093807106990, −8.190604777962639499064596671549, −7.25572705650149307915082549465, −6.82214262234629994428786630148, −5.92643394581920588942276481674, −5.24630393837176712137918176452, −4.30567612608975001316557370422, −3.16723490275137604975258479891, −2.34124668912049794564132669395, −1.62815264106558070896990678800,
0.68625410106270524358785150689, 1.58484315051892057223864877176, 2.54643892949723033796755149978, 3.96681041282961271574511092734, 4.72032870853404723792517997413, 5.47261694272528910534137063033, 5.81875021544196545719147624453, 7.09898674373438442690001547150, 7.966485478736921604171171759847, 8.487654156024115750049990806390