L(s) = 1 | + (−51 + 62.9i)3-s − 1.13e3i·5-s − 3.09e3·7-s + (−1.35e3 − 6.41e3i)9-s − 1.13e3i·11-s − 7.29e3·13-s + (7.12e4 + 5.77e4i)15-s + 5.89e4i·17-s − 8.03e4·19-s + (1.57e5 − 1.94e5i)21-s − 9.74e4i·23-s − 8.92e5·25-s + (4.73e5 + 2.41e5i)27-s − 8.64e5i·29-s + 4.35e5·31-s + ⋯ |
L(s) = 1 | + (−0.629 + 0.776i)3-s − 1.81i·5-s − 1.28·7-s + (−0.207 − 0.978i)9-s − 0.0773i·11-s − 0.255·13-s + (1.40 + 1.14i)15-s + 0.705i·17-s − 0.616·19-s + (0.811 − 1.00i)21-s − 0.348i·23-s − 2.28·25-s + (0.890 + 0.455i)27-s − 1.22i·29-s + 0.472·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.229603 - 0.481621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.229603 - 0.481621i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (51 - 62.9i)T \) |
good | 5 | \( 1 + 1.13e3iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.09e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.13e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 7.29e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.89e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 8.03e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 9.74e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.64e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 4.35e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.15e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.71e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 9.90e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 6.70e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.00e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.59e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.93e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.80e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.36e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.52e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.34e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.75e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.82e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.95e7T + 7.83e15T^{2} \) |
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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
−17.21482746106720379639862545719, −16.50815603606336456160906881401, −15.49936661717884394921006257882, −13.09158384914558410691953819291, −12.10179778674517631898903364219, −10.01373990657277874261341800955, −8.820442190361542846253672113124, −5.91441747836610582575367729922, −4.25455643394907026497782402673, −0.34886356695735904396350483660,
2.83317309155976541842739968856, 6.27747016969976537408119544359, 7.22031344423970737463721534734, 10.09502317702145436350847680453, 11.40129985347850220984691901489, 13.01539244434248449812613017613, 14.44390812927988331574038749502, 16.07983255575924744869922157749, 17.71891058070302307748402858135, 18.76762627476353375259606061739