L(s) = 1 | − 1.70i·3-s + (−2.17 + 0.539i)5-s + 2.63i·7-s + 0.0783·9-s + 5.41·11-s − 6.34i·13-s + (0.921 + 3.70i)15-s − 3.41i·17-s − 3.26·19-s + 4.49·21-s − 1.36i·23-s + (4.41 − 2.34i)25-s − 5.26i·27-s + 2·29-s + 4.68·31-s + ⋯ |
L(s) = 1 | − 0.986i·3-s + (−0.970 + 0.241i)5-s + 0.994i·7-s + 0.0261·9-s + 1.63·11-s − 1.75i·13-s + (0.237 + 0.957i)15-s − 0.829i·17-s − 0.748·19-s + 0.981·21-s − 0.285i·23-s + (0.883 − 0.468i)25-s − 1.01i·27-s + 0.371·29-s + 0.840·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04013 - 0.813323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04013 - 0.813323i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.17 - 0.539i)T \) |
good | 3 | \( 1 + 1.70iT - 3T^{2} \) |
| 7 | \( 1 - 2.63iT - 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 6.34iT - 13T^{2} \) |
| 17 | \( 1 + 3.41iT - 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 1.36iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 + 5.75iT - 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 4.44iT - 43T^{2} \) |
| 47 | \( 1 - 4.78iT - 47T^{2} \) |
| 53 | \( 1 + 1.65iT - 53T^{2} \) |
| 59 | \( 1 - 3.26T + 59T^{2} \) |
| 61 | \( 1 - 2.49T + 61T^{2} \) |
| 67 | \( 1 + 7.86iT - 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 + 13.5iT - 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 14.9iT - 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 - 4.58iT - 97T^{2} \) |
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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
−10.52600522759019618495167719152, −9.366716494333968171501735006047, −8.409792739317479390981251361498, −7.80829434976398134761504548372, −6.80260939939804957434220099178, −6.18040911176930181377705054361, −4.85867903245936313465846955545, −3.59224656910736929562405462684, −2.45357920631419073621964914111, −0.837512602725564302670678358522,
1.43111198691620754670780777606, 3.80636356463802694250025123771, 4.00376065829358913741521227302, 4.76958914249168510742625634764, 6.58624150239615852890033729991, 7.00745982763725617832567018472, 8.381009967670667420616703259120, 9.041747066856190155060571414042, 9.927208761517127981215231072036, 10.70279657720682617389374531513