| L(s) = 1 | + (45.1 − 12.2i)3-s − 464.·5-s − 639. i·7-s + (1.88e3 − 1.10e3i)9-s + 1.35e3i·11-s + 1.04e4i·13-s + (−2.09e4 + 5.69e3i)15-s − 2.64e4i·17-s − 1.84e4·19-s + (−7.83e3 − 2.88e4i)21-s − 8.61e4·23-s + 1.37e5·25-s + (7.15e4 − 7.31e4i)27-s + 5.92e4·29-s + 1.22e5i·31-s + ⋯ |
| L(s) = 1 | + (0.964 − 0.262i)3-s − 1.66·5-s − 0.704i·7-s + (0.862 − 0.506i)9-s + 0.306i·11-s + 1.31i·13-s + (−1.60 + 0.435i)15-s − 1.30i·17-s − 0.617·19-s + (−0.184 − 0.679i)21-s − 1.47·23-s + 1.75·25-s + (0.699 − 0.714i)27-s + 0.450·29-s + 0.740i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.439147904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.439147904\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-45.1 + 12.2i)T \) |
| good | 5 | \( 1 + 464.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 639. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.35e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.04e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.64e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.84e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.61e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.92e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.22e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 8.41e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 8.14e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 4.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.46e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.35e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.80e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 3.61e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.92e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.55e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 5.74e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 9.67e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 3.10e6T + 8.07e13T^{2} \) |
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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
−11.69856596313222791124064582831, −10.39954586421402887607220176017, −9.220827608561944094014953849990, −8.258744680003997518268833312354, −7.40922494461909035282883556524, −6.78353591733672727797018904913, −4.40170130795862048521018248087, −3.98262858235679268964492130201, −2.61301244611640778676774716668, −1.01278241298516801242488438511,
0.37764550133853541197996757007, 2.27548801227273665786089046160, 3.54028992585646571177089063101, 4.19471155576636379769797128251, 5.77013000665978500353447610640, 7.40719086479304835122971775000, 8.247283424926310176508355925126, 8.637851653232989368861532113155, 10.16074427144820147926213692624, 10.99955287094813361156546261037
