L(s) = 1 | − 2-s + i·3-s + 4-s + (−0.623 − 2.14i)5-s − i·6-s − 5.25i·7-s − 8-s − 9-s + (0.623 + 2.14i)10-s + 1.83·11-s + i·12-s − 2.67·13-s + 5.25i·14-s + (2.14 − 0.623i)15-s + 16-s − 2.30·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.278 − 0.960i)5-s − 0.408i·6-s − 1.98i·7-s − 0.353·8-s − 0.333·9-s + (0.197 + 0.679i)10-s + 0.551·11-s + 0.288i·12-s − 0.743·13-s + 1.40i·14-s + (0.554 − 0.161i)15-s + 0.250·16-s − 0.559·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4444966055\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4444966055\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.623 + 2.14i)T \) |
| 37 | \( 1 + (-6.05 - 0.627i)T \) |
good | 7 | \( 1 + 5.25iT - 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 + 5.44iT - 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 4.77iT - 29T^{2} \) |
| 31 | \( 1 - 7.06iT - 31T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 6.72T + 43T^{2} \) |
| 47 | \( 1 - 8.67iT - 47T^{2} \) |
| 53 | \( 1 - 13.4iT - 53T^{2} \) |
| 59 | \( 1 - 5.76iT - 59T^{2} \) |
| 61 | \( 1 + 8.95iT - 61T^{2} \) |
| 67 | \( 1 - 1.91iT - 67T^{2} \) |
| 71 | \( 1 + 3.15T + 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 + 16.1iT - 79T^{2} \) |
| 83 | \( 1 + 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 4.74T + 97T^{2} \) |
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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
−9.414276361694485268354849272661, −8.832729212412564162998921165776, −7.77807757719480058466415517328, −7.23197383330578708598437408190, −6.26936685873064456906010037853, −4.72229915555649906611512548372, −4.38935666663568461133436845853, −3.22871405304752919344450385803, −1.43172669634978141621169806872, −0.24479198121578708173206726299,
2.09374174304032115427337794998, 2.48597123871234669319907582556, 3.84850246296212135023535320882, 5.55027267705064857326389755851, 6.17795960827957318606011896307, 6.88331545484737420155244145320, 8.072480846225231983933789081004, 8.293494219059097338283828982077, 9.565473201456256210861484292320, 9.885485396045724646944263996466