| L(s) = 1 | + (1.5 + 0.866i)3-s + (3.79 − 2.19i)5-s + (−0.5 + 6.98i)7-s + (1.5 + 2.59i)9-s + (−9.79 + 16.9i)11-s + 6.11i·13-s + 7.59·15-s + (7.59 + 4.38i)17-s + (26.2 − 15.1i)19-s + (−6.79 + 10.0i)21-s + (−12 − 20.7i)23-s + (−2.89 + 5.00i)25-s + 5.19i·27-s + 13.5·29-s + (24.2 + 14.0i)31-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.288i)3-s + (0.759 − 0.438i)5-s + (−0.0714 + 0.997i)7-s + (0.166 + 0.288i)9-s + (−0.890 + 1.54i)11-s + 0.470i·13-s + 0.506·15-s + (0.446 + 0.257i)17-s + (1.38 − 0.799i)19-s + (−0.323 + 0.478i)21-s + (−0.521 − 0.903i)23-s + (−0.115 + 0.200i)25-s + 0.192i·27-s + 0.468·29-s + (0.783 + 0.452i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.77940 + 1.09397i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77940 + 1.09397i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 6.98i)T \) |
| good | 5 | \( 1 + (-3.79 + 2.19i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (9.79 - 16.9i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 6.11iT - 169T^{2} \) |
| 17 | \( 1 + (-7.59 - 4.38i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-26.2 + 15.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (12 + 20.7i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 13.5T + 841T^{2} \) |
| 31 | \( 1 + (-24.2 - 14.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-24.2 - 42.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 7.14iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-34.5 + 19.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-30.7 + 53.3i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (66.7 + 38.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-0.373 + 0.215i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (28.4 - 49.3i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 123.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-31.0 - 17.9i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (26.0 + 45.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 136. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-13.2 + 7.63i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 34.1iT - 9.40e3T^{2} \) |
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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
−11.66930151945708726795099705760, −10.11670427785512333076494714501, −9.743124731681961182339856195119, −8.828905580781021321608355946281, −7.87083247624874158815409708685, −6.67130131699729184297960807919, −5.34508486440339797792391941829, −4.68722408545423394794943124187, −2.88373273893957534673896632401, −1.86237948458439677149166284284,
0.974639040339644966956502793960, 2.75132306997334232453600966856, 3.66054453874523256520451581483, 5.44124935971378687724210735372, 6.24730703965959006430259467931, 7.60428099855757764927401932316, 8.058547506993196235502661599235, 9.498962138817133696288515686513, 10.20370178891042819434534121400, 10.98476440043763958625574962457
