L(s) = 1 | + i·3-s + i·5-s − 2.13·7-s − 9-s + 4.16·11-s + 2.81·13-s − 15-s − 1.82i·17-s − 1.32·19-s − 2.13i·21-s + (4.63 − 1.24i)23-s − 25-s − i·27-s − 4.92·29-s − 10.5i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 0.806·7-s − 0.333·9-s + 1.25·11-s + 0.781·13-s − 0.258·15-s − 0.441i·17-s − 0.304·19-s − 0.465i·21-s + (0.965 − 0.259i)23-s − 0.200·25-s − 0.192i·27-s − 0.914·29-s − 1.89i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.408822872\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.408822872\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-4.63 + 1.24i)T \) |
good | 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 + 1.82iT - 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + 10.5iT - 31T^{2} \) |
| 37 | \( 1 + 4.95iT - 37T^{2} \) |
| 41 | \( 1 - 0.656T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 + 7.28iT - 47T^{2} \) |
| 53 | \( 1 - 7.24iT - 53T^{2} \) |
| 59 | \( 1 + 8.42iT - 59T^{2} \) |
| 61 | \( 1 + 3.54iT - 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 0.897iT - 71T^{2} \) |
| 73 | \( 1 - 4.48T + 73T^{2} \) |
| 79 | \( 1 - 7.73T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 + 5.21iT - 89T^{2} \) |
| 97 | \( 1 + 0.00964iT - 97T^{2} \) |
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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
−8.120697373995618468918992845842, −7.23104770337914968536527151293, −6.51414430225826313392436057856, −6.08254761107419768718854467585, −5.18763512103533473551600237585, −4.14343678015387385109684625302, −3.66153165657785056074435027467, −2.92269601158444482779788967941, −1.80827743079938598770305685489, −0.40330832460303744368047953399,
1.09987119534717291911170221713, 1.70085648036335777675131237635, 3.10930747215358260252581609990, 3.61122704567370293033170037938, 4.58279056358412884270852090824, 5.47479506211421722172159873430, 6.35397418825681122708480081437, 6.64373797757489416903190882330, 7.42325112652955604432047550388, 8.427080956081786981490172623329