L(s) = 1 | + 1.29i·2-s + (−2.20 − 2.03i)3-s + 2.31·4-s + 5.31i·5-s + (2.64 − 2.86i)6-s + 12.0·7-s + 8.19i·8-s + (0.718 + 8.97i)9-s − 6.89·10-s − 11.3i·11-s + (−5.10 − 4.71i)12-s − 10.1·13-s + 15.6i·14-s + (10.8 − 11.7i)15-s − 1.36·16-s − 4.40i·17-s + ⋯ |
L(s) = 1 | + 0.648i·2-s + (−0.734 − 0.678i)3-s + 0.579·4-s + 1.06i·5-s + (0.440 − 0.476i)6-s + 1.71·7-s + 1.02i·8-s + (0.0798 + 0.996i)9-s − 0.689·10-s − 1.03i·11-s + (−0.425 − 0.392i)12-s − 0.779·13-s + 1.11i·14-s + (0.720 − 0.780i)15-s − 0.0854·16-s − 0.258i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15635 + 0.506270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15635 + 0.506270i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.20 + 2.03i)T \) |
| 23 | \( 1 + 4.79iT \) |
good | 2 | \( 1 - 1.29iT - 4T^{2} \) |
| 5 | \( 1 - 5.31iT - 25T^{2} \) |
| 7 | \( 1 - 12.0T + 49T^{2} \) |
| 11 | \( 1 + 11.3iT - 121T^{2} \) |
| 13 | \( 1 + 10.1T + 169T^{2} \) |
| 17 | \( 1 + 4.40iT - 289T^{2} \) |
| 19 | \( 1 + 14.8T + 361T^{2} \) |
| 29 | \( 1 + 33.0iT - 841T^{2} \) |
| 31 | \( 1 + 22.9T + 961T^{2} \) |
| 37 | \( 1 + 42.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 33.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 74.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 22.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 96.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 126.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 65.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 22.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 37.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 73.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 72.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 0.728T + 9.40e3T^{2} \) |
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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
−14.54577372203426354691008748580, −14.00422309422518919443385515944, −12.15686829870793590243417169570, −11.09349893069955809187612861244, −10.83800049803167233055233198487, −8.214622161261227533955886520496, −7.41011670871996816480744230803, −6.33484512817050293705068378866, −5.14072250196282626190761151168, −2.21303242629458173859764704959,
1.63464243621635988174400551265, 4.35468980641529923293689222758, 5.26274209118153803820922542001, 7.20760682299593594317724354143, 8.800894231029519441526828394246, 10.15941652463104302838231787317, 11.07000041245280216261035341265, 12.05533215150293078716556195785, 12.63386440639310775875888354340, 14.66836127487628913244502210630