L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 4.41i·5-s + 3.24·7-s + 2.82i·8-s + 6.24·10-s + 0.171i·11-s − 4.58i·14-s + 4.00·16-s − 8.82i·20-s + 0.242·22-s − 14.4·25-s − 6.48·28-s − 2.82i·29-s + 9.24·31-s − 5.65i·32-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 1.97i·5-s + 1.22·7-s + 1.00i·8-s + 1.97·10-s + 0.0517i·11-s − 1.22i·14-s + 1.00·16-s − 1.97i·20-s + 0.0517·22-s − 2.89·25-s − 1.22·28-s − 0.525i·29-s + 1.66·31-s − 1.00i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18774\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18774\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4.41iT - 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 0.171iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 4.07iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 17.8iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−11.80774928306062185451101436052, −11.37522714816070148754386847285, −10.52979752939071557861302780712, −9.864665624649668808362080951209, −8.353079900643625565664156056767, −7.43320751867388141567271288795, −6.08030678483194142331386839723, −4.56654284062922243129795878754, −3.23425853026846556284851884959, −2.09347155128834804444612063635,
1.20357386349153630305809421009, 4.32242328711747486569045006635, 4.93396269287804346789627372966, 5.87698713383913919953825076736, 7.54840850903100823415266873109, 8.406152479796825356247832892286, 8.890519749902944527139694254697, 10.03784094777707466947767850714, 11.65210918947207506448662675421, 12.48815097749150866335251596175