| L(s) = 1 | + (−2.58 + 1.52i)3-s − 2.23i·5-s − 7.48·7-s + (4.32 − 7.89i)9-s + 8.48i·11-s + 10·13-s + (3.41 + 5.77i)15-s + 30.3i·17-s − 26.9·19-s + (19.3 − 11.4i)21-s − 9.17i·23-s − 5.00·25-s + (0.905 + 26.9i)27-s − 26.8i·29-s + 8·31-s + ⋯ |
| L(s) = 1 | + (−0.860 + 0.509i)3-s − 0.447i·5-s − 1.06·7-s + (0.480 − 0.876i)9-s + 0.771i·11-s + 0.769·13-s + (0.227 + 0.384i)15-s + 1.78i·17-s − 1.41·19-s + (0.920 − 0.545i)21-s − 0.398i·23-s − 0.200·25-s + (0.0335 + 0.999i)27-s − 0.925i·29-s + 0.258·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7200604339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7200604339\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.58 - 1.52i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 + 7.48T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 - 30.3iT - 289T^{2} \) |
| 19 | \( 1 + 26.9T + 361T^{2} \) |
| 23 | \( 1 + 9.17iT - 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 - 8T + 961T^{2} \) |
| 37 | \( 1 + 15.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 47.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 14.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 45.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 15.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 87.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 46.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 26.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 36.0T + 9.40e3T^{2} \) |
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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
−9.930395423710826836957118802273, −8.944001180288599820397344553522, −8.207682721126455406592990316786, −6.73139987373445635569560730399, −6.31024726742822420446495427145, −5.42543478516021894489939448820, −4.20897859762914519243283054979, −3.73071082408487159052578083893, −1.94530454952447337104379214566, −0.33920420692028983740464589918,
0.890623570817055792729584255180, 2.53257260387875860482597635374, 3.55571761930710043452182731966, 4.85394196153726604527784551808, 5.91075805426424680843108144372, 6.51722491148253409893380055712, 7.14896105831269141946058469225, 8.219434747881252017701918787118, 9.224749763161881061658353436142, 10.08802950115619562165564106769
