L(s) = 1 | + (−1.58 − 1.22i)2-s + (1.00 + 3.87i)4-s + (−0.540 + 0.935i)5-s + (−5.20 + 3.00i)7-s + (3.16 − 7.34i)8-s + (2 − 0.817i)10-s + (15.3 − 8.86i)11-s + (−6.20 + 10.7i)13-s + (11.9 + 1.62i)14-s + (−13.9 + 7.74i)16-s − 26.3·17-s − 5.19i·19-s + (−4.16 − 1.15i)20-s + (−35.1 − 4.78i)22-s + (−25.8 − 14.9i)23-s + ⋯ |
L(s) = 1 | + (−0.790 − 0.612i)2-s + (0.250 + 0.968i)4-s + (−0.108 + 0.187i)5-s + (−0.744 + 0.429i)7-s + (0.395 − 0.918i)8-s + (0.200 − 0.0817i)10-s + (1.39 − 0.805i)11-s + (−0.477 + 0.827i)13-s + (0.851 + 0.116i)14-s + (−0.874 + 0.484i)16-s − 1.55·17-s − 0.273i·19-s + (−0.208 − 0.0578i)20-s + (−1.59 − 0.217i)22-s + (−1.12 − 0.648i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0393988 + 0.122998i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0393988 + 0.122998i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.58 + 1.22i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.540 - 0.935i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (5.20 - 3.00i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.3 + 8.86i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (6.20 - 10.7i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 26.3T + 289T^{2} \) |
| 19 | \( 1 + 5.19iT - 361T^{2} \) |
| 23 | \( 1 + (25.8 + 14.9i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (2.16 + 3.74i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (38.8 + 22.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 20.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (29.6 - 51.3i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (16.5 - 9.57i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (35.5 - 20.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 70.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-25.0 - 14.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (9.20 + 15.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-82.1 - 47.4i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 83.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 55.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-35.5 + 20.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (30.7 - 17.7i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 26.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + (38.1 + 66.1i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.48201488524114991966693473854, −11.06023196706253581602389137289, −9.627044652709455112826656841312, −9.196979541413591578454476444491, −8.314481044417172058620496324776, −6.88035192170873540929323841825, −6.35070167739087615384612346206, −4.33507828092989860693846147173, −3.25834627777227677426081198121, −1.89290954619906552714000529828,
0.07516104226426905369482547064, 1.87289409834048051288783672264, 3.86055112355454570571666952679, 5.16739956009546575745971988509, 6.55514263992407796250520797226, 6.99889344758972817489359068781, 8.224850031901872459233626621738, 9.179840683080561878116088294126, 9.890833177254267259293123249011, 10.72633748659240428888110628457