L(s) = 1 | + (1.08 + 45.2i)2-s + (399. + 132. i)3-s + (−2.04e3 + 98.5i)4-s + 4.15e3i·5-s + (−5.56e3 + 1.82e4i)6-s + 3.91e4i·7-s + (−6.68e3 − 9.24e4i)8-s + (1.41e5 + 1.05e5i)9-s + (−1.87e5 + 4.51e3i)10-s − 7.15e5·11-s + (−8.30e5 − 2.32e5i)12-s − 6.05e5·13-s + (−1.77e6 + 4.26e4i)14-s + (−5.50e5 + 1.65e6i)15-s + (4.17e6 − 4.03e5i)16-s − 5.31e6i·17-s + ⋯ |
L(s) = 1 | + (0.0240 + 0.999i)2-s + (0.949 + 0.315i)3-s + (−0.998 + 0.0481i)4-s + 0.593i·5-s + (−0.292 + 0.956i)6-s + 0.880i·7-s + (−0.0721 − 0.997i)8-s + (0.801 + 0.598i)9-s + (−0.593 + 0.0142i)10-s − 1.33·11-s + (−0.963 − 0.269i)12-s − 0.452·13-s + (−0.880 + 0.0211i)14-s + (−0.187 + 0.563i)15-s + (0.995 − 0.0960i)16-s − 0.908i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.243884 + 1.77876i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.243884 + 1.77876i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.08 - 45.2i)T \) |
| 3 | \( 1 + (-399. - 132. i)T \) |
good | 5 | \( 1 - 4.15e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 - 3.91e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + 7.15e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 6.05e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.31e6iT - 3.42e13T^{2} \) |
| 19 | \( 1 - 1.51e7iT - 1.16e14T^{2} \) |
| 23 | \( 1 - 4.01e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.84e6iT - 1.22e16T^{2} \) |
| 31 | \( 1 - 2.41e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 - 4.80e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.82e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + 1.12e9iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 4.80e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.89e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 3.77e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.72e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.69e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 1.32e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.00e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.23e9iT - 7.47e20T^{2} \) |
| 83 | \( 1 - 3.04e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 3.87e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 - 3.29e10T + 7.15e21T^{2} \) |
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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
−18.27629679821358910111426184313, −16.24868573699032275434471280272, −15.21240953871893661259948505212, −14.29197580777655521901162739194, −12.82902655534493304006437604002, −10.15192380986184295748353638000, −8.674286357144801528646749969974, −7.31091960407941843840578113346, −5.15868762742027999985473561940, −2.92141650000446132789995840834,
0.817684689479792350279214361980, 2.69358453329266564779330986190, 4.54763785108689033763604066674, 7.77369521371702128682350504447, 9.242980664223395716034289192032, 10.73879374504972438052000678033, 12.83131951190990409972529768554, 13.41419544864453386401557419636, 15.04545964894518452367457076717, 17.14194147918431659445644319962