| L(s) = 1 | + 16.1·3-s − 376. i·5-s + (831. − 363. i)7-s − 1.92e3·9-s − 5.36e3i·11-s − 87.2i·13-s − 6.09e3i·15-s + 1.62e3i·17-s − 8.07e3·19-s + (1.34e4 − 5.88e3i)21-s + 6.73e4i·23-s − 6.36e4·25-s − 6.65e4·27-s − 3.43e4·29-s − 1.39e5·31-s + ⋯ |
| L(s) = 1 | + 0.346·3-s − 1.34i·5-s + (0.916 − 0.400i)7-s − 0.880·9-s − 1.21i·11-s − 0.0110i·13-s − 0.466i·15-s + 0.0800i·17-s − 0.270·19-s + (0.317 − 0.138i)21-s + 1.15i·23-s − 0.814·25-s − 0.650·27-s − 0.261·29-s − 0.843·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.486707643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486707643\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-831. + 363. i)T \) |
| good | 3 | \( 1 - 16.1T + 2.18e3T^{2} \) |
| 5 | \( 1 + 376. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + 5.36e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 87.2iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.62e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 8.07e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.73e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 3.43e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.39e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.89e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.23e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 6.88e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 9.74e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.05e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.16e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 2.11e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 9.31e3iT - 9.09e12T^{2} \) |
| 73 | \( 1 - 6.40e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 7.41e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 1.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.26e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 - 4.31e6iT - 8.07e13T^{2} \) |
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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
−11.69082832659179255921509945588, −10.97849133535972801245005830545, −9.320198970767827631811999257692, −8.511425846069115242399387861202, −7.79897266830837142172954354143, −5.85014152531940486347429373016, −4.89534669039049036842260585390, −3.49287753638253990962459399829, −1.65955854987589151791640192868, −0.40187241965673624759814527637,
2.00589701561855266634961259378, 2.95627561638435441928743919799, 4.59533809878351536654492123046, 6.08816111541606410889485807768, 7.28292751345219643092980573550, 8.277637182155158635449537168470, 9.538557152138866504571272726075, 10.76539599386694865491296260518, 11.45718060058573029025278434129, 12.63462868851196719684500922930
