L(s) = 1 | + (−0.530 − 1.31i)2-s + 3-s + (−1.43 + 1.39i)4-s + 2.85i·5-s + (−0.530 − 1.31i)6-s + 4.91·7-s + (2.58 + 1.14i)8-s + 9-s + (3.74 − 1.51i)10-s − 6.48·11-s + (−1.43 + 1.39i)12-s − 2.89i·13-s + (−2.60 − 6.44i)14-s + 2.85i·15-s + (0.131 − 3.99i)16-s + 4.14·17-s + ⋯ |
L(s) = 1 | + (−0.375 − 0.927i)2-s + 0.577·3-s + (−0.718 + 0.695i)4-s + 1.27i·5-s + (−0.216 − 0.535i)6-s + 1.85·7-s + (0.914 + 0.405i)8-s + 0.333·9-s + (1.18 − 0.478i)10-s − 1.95·11-s + (−0.414 + 0.401i)12-s − 0.802i·13-s + (−0.696 − 1.72i)14-s + 0.737i·15-s + (0.0329 − 0.999i)16-s + 1.00·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62342 + 0.121620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62342 + 0.121620i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.530 + 1.31i)T \) |
| 3 | \( 1 - T \) |
| 67 | \( 1 + (6.66 + 4.75i)T \) |
good | 5 | \( 1 - 2.85iT - 5T^{2} \) |
| 7 | \( 1 - 4.91T + 7T^{2} \) |
| 11 | \( 1 + 6.48T + 11T^{2} \) |
| 13 | \( 1 + 2.89iT - 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 4.77iT - 19T^{2} \) |
| 23 | \( 1 - 6.27iT - 23T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 - 0.791T + 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 - 7.22iT - 41T^{2} \) |
| 43 | \( 1 + 4.34T + 43T^{2} \) |
| 47 | \( 1 - 1.23iT - 47T^{2} \) |
| 53 | \( 1 + 0.123iT - 53T^{2} \) |
| 59 | \( 1 - 2.67iT - 59T^{2} \) |
| 61 | \( 1 + 4.75iT - 61T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 - 0.728T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 0.704T + 89T^{2} \) |
| 97 | \( 1 + 9.89iT - 97T^{2} \) |
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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
−10.42258062249661463256036714186, −9.803008224833294853268863838409, −8.283614100324291927510604746564, −7.87516335408369703443398101831, −7.51757087857360222626817645391, −5.58376274378716017451170661230, −4.76264165543374006312978331360, −3.36089127958555195095682140941, −2.68378796080770454969270083076, −1.57599902424493073151238140548,
0.967106419799678623066359907242, 2.27998593890034866991910230538, 4.48979964268628046552613219341, 4.85522906045869145221338669236, 5.56651950845221702365276046085, 7.16053176229597243037020917548, 7.921651035739545084955885352987, 8.463268409529841783016441470225, 8.888609673112493284466080796405, 10.09717533282652050464056621098