L(s) = 1 | − i·2-s + (−0.0198 + 1.73i)3-s − 4-s + 1.44·5-s + (1.73 + 0.0198i)6-s + (−2.59 + 0.507i)7-s + i·8-s + (−2.99 − 0.0687i)9-s − 1.44i·10-s + 4.91i·11-s + (0.0198 − 1.73i)12-s − i·13-s + (0.507 + 2.59i)14-s + (−0.0287 + 2.50i)15-s + 16-s − 2.13·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.0114 + 0.999i)3-s − 0.5·4-s + 0.647·5-s + (0.707 + 0.00810i)6-s + (−0.981 + 0.191i)7-s + 0.353i·8-s + (−0.999 − 0.0229i)9-s − 0.457i·10-s + 1.48i·11-s + (0.00573 − 0.499i)12-s − 0.277i·13-s + (0.135 + 0.693i)14-s + (−0.00742 + 0.647i)15-s + 0.250·16-s − 0.518·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.597724 + 0.717554i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.597724 + 0.717554i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.0198 - 1.73i)T \) |
| 7 | \( 1 + (2.59 - 0.507i)T \) |
| 13 | \( 1 + iT \) |
good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 11 | \( 1 - 4.91iT - 11T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 - 2.79iT - 19T^{2} \) |
| 23 | \( 1 - 2.25iT - 23T^{2} \) |
| 29 | \( 1 - 1.23iT - 29T^{2} \) |
| 31 | \( 1 - 5.68iT - 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.01T + 41T^{2} \) |
| 43 | \( 1 + 0.729T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 - 4.45iT - 61T^{2} \) |
| 67 | \( 1 - 1.73T + 67T^{2} \) |
| 71 | \( 1 + 7.37iT - 71T^{2} \) |
| 73 | \( 1 + 11.4iT - 73T^{2} \) |
| 79 | \( 1 - 3.64T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 12.6iT - 97T^{2} \) |
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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
−10.53331161101338639787556256385, −10.36627863777741283414859537620, −9.325680922450365819608017039516, −9.070421335976013796674225893418, −7.53531987140526969403793856863, −6.22592255041185161166901612358, −5.30271736148185607692167836334, −4.27184195497167697432791620343, −3.23747728783795534323484599572, −2.07362362642906402154511107355,
0.51718520434476939517866716432, 2.42478891236945411618938932363, 3.70771863063760100278595091592, 5.41012774009987068166035287337, 6.20279814861314079609486820121, 6.70274561382754140250200548393, 7.74662433861446457983233008885, 8.752156735657042088960702888543, 9.333872846837214861664131529647, 10.54242212447140716668073297814