L(s) = 1 | + 4i·2-s + (14.2 − 6.20i)3-s − 16·4-s + 97.9·5-s + (24.8 + 57.1i)6-s − 18.8i·7-s − 64i·8-s + (165. − 177. i)9-s + 391. i·10-s + 79.1·11-s + (−228. + 99.3i)12-s + 704.·13-s + 75.3·14-s + (1.40e3 − 608. i)15-s + 256·16-s − 2.09e3·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.917 − 0.398i)3-s − 0.5·4-s + 1.75·5-s + (0.281 + 0.648i)6-s − 0.145i·7-s − 0.353i·8-s + (0.682 − 0.730i)9-s + 1.23i·10-s + 0.197·11-s + (−0.458 + 0.199i)12-s + 1.15·13-s + 0.102·14-s + (1.60 − 0.697i)15-s + 0.250·16-s − 1.76·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.491884814\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.491884814\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 + (-14.2 + 6.20i)T \) |
| 23 | \( 1 + (392. - 2.50e3i)T \) |
good | 5 | \( 1 - 97.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 18.8iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 79.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 704.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.64e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 + 7.95e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.44e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.22e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.11e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.01e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.68e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.99e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.83e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.11e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 1.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.22e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 4.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5iT - 8.58e9T^{2} \) |
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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
−13.11923570451290402109104494951, −11.16830888215657448168827628011, −9.697753334280117489070580042552, −9.189482171424983190451445320297, −8.150199131896848098103045273841, −6.65479441216203450487278700423, −6.11931029659670564531020775983, −4.44797916201638997002395675410, −2.63553256646723988808093023187, −1.31845299394596289769774053872,
1.60140860739724940527164452775, 2.46025069213617803304310774394, 3.93686757621606406793269107226, 5.38172096478263786625035988583, 6.67598683672731401683156160006, 8.786435072795879449316159054958, 8.930186943276802382185869955966, 10.30797392627527968485779464253, 10.69465332209361119604335942465, 12.47707737027902525510005112772