L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − i·5-s − 0.999i·8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73·17-s − 1.73i·19-s + (0.866 − 0.499i)20-s − 1.73·23-s − 25-s + 31-s + (0.866 − 0.499i)32-s + (−1.49 − 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − i·5-s − 0.999i·8-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)16-s + 1.73·17-s − 1.73i·19-s + (0.866 − 0.499i)20-s − 1.73·23-s − 25-s + 31-s + (0.866 − 0.499i)32-s + (−1.49 − 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6819262146\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6819262146\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - 1.73T + T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + 1.73T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.73iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.779681599709291891197608617088, −9.171085672457500616098977651728, −8.242405793562806391399728382769, −7.78951021353045043087839930429, −6.69218734271625664329966040459, −5.59601600180493254397855074635, −4.51567241286491564142400744700, −3.47105853037303369137181192217, −2.21923413693686383482525452235, −0.893866419902744668565386984301,
1.60710571489152586030230424077, 2.88950496674166066815577695259, 4.06366637854581671650160072314, 5.78487945793853369233458768088, 5.95125536030593633412552129607, 7.16960425272490589735042832760, 7.81188221421039848535023044515, 8.397692781226793200637204925531, 9.748336323406772493099518654573, 10.10405513083396779112910541810