L(s) = 1 | + (0.5 + 0.866i)3-s + 0.267i·5-s + (−0.633 − 0.366i)7-s + (−0.499 + 0.866i)9-s + (4.09 − 2.36i)11-s + (2.59 + 2.5i)13-s + (−0.232 + 0.133i)15-s + (−1.13 + 1.96i)17-s + (1.09 + 0.633i)19-s − 0.732i·21-s + (3.09 + 5.36i)23-s + 4.92·25-s − 0.999·27-s + (−1.23 − 2.13i)29-s + 5.46i·31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + 0.119i·5-s + (−0.239 − 0.138i)7-s + (−0.166 + 0.288i)9-s + (1.23 − 0.713i)11-s + (0.720 + 0.693i)13-s + (−0.0599 + 0.0345i)15-s + (−0.275 + 0.476i)17-s + (0.251 + 0.145i)19-s − 0.159i·21-s + (0.645 + 1.11i)23-s + 0.985·25-s − 0.192·27-s + (−0.228 − 0.396i)29-s + 0.981i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57933 + 0.660123i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57933 + 0.660123i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-2.59 - 2.5i)T \) |
good | 5 | \( 1 - 0.267iT - 5T^{2} \) |
| 7 | \( 1 + (0.633 + 0.366i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.13 - 1.96i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 0.633i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.09 - 5.36i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.23 + 2.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (9.06 - 5.23i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.86 + 5.69i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.83 - 6.63i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8.19iT - 47T^{2} \) |
| 53 | \( 1 - 0.464T + 53T^{2} \) |
| 59 | \( 1 + (6.92 + 4i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.598 - 1.03i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.63 + 5.56i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.09 - 0.633i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.73iT - 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + (-2.19 + 1.26i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.19 - 3i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.81567610084212723285475585461, −9.727838671998073031428649936489, −8.977812883619283936323426681445, −8.391105197685961634187259587762, −7.01692139927745341119979734936, −6.32404254896505002427804470460, −5.15462137014560482086804505998, −3.90299633300606398718795154443, −3.27748738744588351168379374794, −1.49182821399345518166288024057,
1.09935040662624855569028054693, 2.57995845748586323648566589504, 3.76807867685129930061042082637, 4.93175656160080098424383277802, 6.20525699269522455494798904006, 6.89140201592224606500353821929, 7.82420297013358269313464485386, 8.945581041985939672453921832613, 9.310122639957589206313806802661, 10.57677794781225045111296256171