L(s) = 1 | + (1.54 + 2.67i)2-s + (−4.5 + 7.79i)3-s + (11.2 − 19.4i)4-s + (−6.89 − 11.9i)5-s − 27.8·6-s + 168.·8-s + (−40.5 − 70.1i)9-s + (21.3 − 36.9i)10-s + (1.83 − 3.17i)11-s + (100. + 174. i)12-s − 780.·13-s + 124.·15-s + (−98.8 − 171. i)16-s + (−25.0 + 43.3i)17-s + (125. − 216. i)18-s + (−531. − 920. i)19-s + ⋯ |
L(s) = 1 | + (0.273 + 0.473i)2-s + (−0.288 + 0.499i)3-s + (0.350 − 0.607i)4-s + (−0.123 − 0.213i)5-s − 0.315·6-s + 0.929·8-s + (−0.166 − 0.288i)9-s + (0.0674 − 0.116i)10-s + (0.00456 − 0.00791i)11-s + (0.202 + 0.350i)12-s − 1.28·13-s + 0.142·15-s + (−0.0965 − 0.167i)16-s + (−0.0210 + 0.0363i)17-s + (0.0910 − 0.157i)18-s + (−0.337 − 0.585i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.269773703\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.269773703\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.54 - 2.67i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (6.89 + 11.9i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-1.83 + 3.17i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 780.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (25.0 - 43.3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (531. + 920. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.05e3 + 3.55e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 1.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (2.75e3 - 4.78e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.07e3 + 5.32e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.76e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.47e4 + 2.55e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-9.62e3 + 1.66e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (3.30e3 - 5.73e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.83e4 + 3.18e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.34e4 - 4.06e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 4.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.47e4 - 2.55e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.10e4 + 1.91e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 3.89e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.02e4 + 1.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.77e4T + 8.58e9T^{2} \) |
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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
−11.89837666039436610753212056180, −10.71748805495290615541570289604, −10.06164554920929649365110187462, −8.823380543755808115024585209302, −7.39120932916551362209199690972, −6.37592711351547245764752747362, −5.20066754143186533046198425737, −4.34442500230938741191871968561, −2.32268495874871532115886534374, −0.38197842950764457984267666476,
1.70875169586454314612705475455, 2.95892122442717097740966620336, 4.34820668800424078627033154184, 5.85486199494106596798094652277, 7.31476707754188228629466641364, 7.77167114068840606292140538132, 9.424190380004572070381880940177, 10.68735350549158935835144808650, 11.57007486370243428607757930070, 12.32573954201946914776331483726