L(s) = 1 | − 1.41·2-s − 1.73i·3-s + 2.00·4-s − 2.23i·5-s + 2.44i·6-s + (−6.70 − 2.02i)7-s − 2.82·8-s − 2.99·9-s + 3.16i·10-s − 5.00·11-s − 3.46i·12-s + 9.06i·13-s + (9.47 + 2.86i)14-s − 3.87·15-s + 4.00·16-s − 19.7i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s − 0.447i·5-s + 0.408i·6-s + (−0.957 − 0.288i)7-s − 0.353·8-s − 0.333·9-s + 0.316i·10-s − 0.455·11-s − 0.288i·12-s + 0.697i·13-s + (0.676 + 0.204i)14-s − 0.258·15-s + 0.250·16-s − 1.16i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0285923 + 0.193684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0285923 + 0.193684i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (6.70 + 2.02i)T \) |
good | 11 | \( 1 + 5.00T + 121T^{2} \) |
| 13 | \( 1 - 9.06iT - 169T^{2} \) |
| 17 | \( 1 + 19.7iT - 289T^{2} \) |
| 19 | \( 1 - 25.6iT - 361T^{2} \) |
| 23 | \( 1 + 40.9T + 529T^{2} \) |
| 29 | \( 1 + 22.3T + 841T^{2} \) |
| 31 | \( 1 - 15.9iT - 961T^{2} \) |
| 37 | \( 1 + 44.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 27.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 30.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 65.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 32.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 83.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 72.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 67.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 30.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 72.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 113. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 103. iT - 9.40e3T^{2} \) |
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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
−11.81546914520498869738069075759, −10.39409146712502877505938206925, −9.644615279544272350571113881618, −8.610903406957405007410591894417, −7.58175549795877651929007416857, −6.66927837949969088599758732479, −5.53154024513644878697823257672, −3.65876682380825580706560970191, −1.96947072880861016113678569573, −0.12726194643307457236450208136,
2.51754508473643973126804886287, 3.76158279374081791971704362091, 5.58691772237443574103303752661, 6.54695282392107001394671217423, 7.82184864898721886492414522022, 8.838988910525826823008023210113, 9.899065920066564533147646317387, 10.43224968155618424721629027866, 11.43855214022334836499109678235, 12.57914539149010603216612122901