| L(s) = 1 | − i·2-s − 4-s − 5-s + (0.0951 + 2.64i)7-s + i·8-s + i·10-s − 5.28i·11-s − 2.19i·13-s + (2.64 − 0.0951i)14-s + 16-s + 1.04·17-s − 6.43i·19-s + 20-s − 5.28·22-s − 7.47i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447·5-s + (0.0359 + 0.999i)7-s + 0.353i·8-s + 0.316i·10-s − 1.59i·11-s − 0.607i·13-s + (0.706 − 0.0254i)14-s + 0.250·16-s + 0.253·17-s − 1.47i·19-s + 0.223·20-s − 1.12·22-s − 1.55i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.501506 - 0.927563i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.501506 - 0.927563i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.0951 - 2.64i)T \) |
| good | 11 | \( 1 + 5.28iT - 11T^{2} \) |
| 13 | \( 1 + 2.19iT - 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 + 6.43iT - 19T^{2} \) |
| 23 | \( 1 + 7.47iT - 23T^{2} \) |
| 29 | \( 1 + 7.47iT - 29T^{2} \) |
| 31 | \( 1 - 9.09iT - 31T^{2} \) |
| 37 | \( 1 + 0.855T + 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 + 0.954T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 3.09iT - 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 8.05iT - 61T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 - 6.43iT - 71T^{2} \) |
| 73 | \( 1 + 4.57iT - 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 4.38T + 83T^{2} \) |
| 89 | \( 1 + 4.28T + 89T^{2} \) |
| 97 | \( 1 - 11.8iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.63245892449269262124813157206, −9.327423894370249965411020894520, −8.624930495207706022931171028632, −8.055264234126454044889623726036, −6.58268986725538445615267625553, −5.60881974478670581297570903430, −4.65610098328064205502335816571, −3.29527547884879116520987595398, −2.55824511298888357705544286123, −0.60619823231526538509081284876,
1.60761816954736093413909103462, 3.67052015898075259734340120727, 4.35355590436271857732303902329, 5.42475118598656336624505263003, 6.65602831224626124036153828677, 7.48057088512620221383927999821, 7.83875994877388881390425979040, 9.225714035218589971537542661146, 9.888322112958456355415363967745, 10.72565824892128268903805128763
