L(s) = 1 | + (−2.72 − 0.772i)2-s + (6.80 + 4.20i)4-s − 3.33i·5-s + 16.9i·7-s + (−15.2 − 16.6i)8-s + (−2.57 + 9.06i)10-s − 16.8·11-s + 25.0·13-s + (13.0 − 45.9i)14-s + (28.6 + 57.2i)16-s + 116. i·17-s + 85.4i·19-s + (14.0 − 22.6i)20-s + (45.7 + 13.0i)22-s + 158.·23-s + ⋯ |
L(s) = 1 | + (−0.961 − 0.273i)2-s + (0.850 + 0.525i)4-s − 0.297i·5-s + 0.912i·7-s + (−0.674 − 0.737i)8-s + (−0.0813 + 0.286i)10-s − 0.461·11-s + 0.535·13-s + (0.249 − 0.877i)14-s + (0.447 + 0.894i)16-s + 1.66i·17-s + 1.03i·19-s + (0.156 − 0.253i)20-s + (0.443 + 0.126i)22-s + 1.43·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.782881 + 0.436617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.782881 + 0.436617i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.72 + 0.772i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.33iT - 125T^{2} \) |
| 7 | \( 1 - 16.9iT - 343T^{2} \) |
| 11 | \( 1 + 16.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 25.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 116. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 85.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 36.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 353.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 144. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 368. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 96.0iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 294.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 301. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 312.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 602. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 856. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 218.T + 9.12e5T^{2} \) |
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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
−12.88173156146921259005210082016, −12.37808413173628357637475485060, −11.06142891064106441812153610087, −10.24993933729060766147362556168, −8.826080840488870352759260005454, −8.382359359986771037012460607086, −6.83604676861352659327123452745, −5.50130964753293465178669971259, −3.34629416603008498227498878650, −1.61712412330441366289475659958,
0.70683793051078551320981593027, 2.86202390242230195024421659298, 5.02553571407447225969326975812, 6.72590802678130999495595801274, 7.41011055341046585734367917318, 8.735378335856488336128252027957, 9.808986567833794513515697316799, 10.84396794327313272470286371617, 11.53223524118579545405754092814, 13.21194992123577017025230816826