L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + i·5-s + (−0.866 + 0.499i)6-s + (4.02 − 2.32i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (3.81 + 2.20i)11-s − 0.999·12-s + (−3.35 − 1.32i)13-s + 4.64·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + 0.447i·5-s + (−0.353 + 0.204i)6-s + (1.52 − 0.877i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (1.14 + 0.663i)11-s − 0.288·12-s + (−0.930 − 0.366i)13-s + 1.24·14-s + (−0.223 − 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62799 + 1.12348i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62799 + 1.12348i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (3.35 + 1.32i)T \) |
good | 7 | \( 1 + (-4.02 + 2.32i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.81 - 2.20i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.96 - 4.02i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.488 - 0.845i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.15 + 3.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.44iT - 31T^{2} \) |
| 37 | \( 1 + (-3.28 - 1.89i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.31 + 3.64i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.358 - 0.620i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.75iT - 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + (1.88 - 1.09i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.73 + 6.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.58 + 0.912i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.88 - 3.97i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4.36iT - 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 3.51iT - 83T^{2} \) |
| 89 | \( 1 + (-7.07 - 4.08i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.9 + 6.91i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.62919227687176000370831493050, −10.56156504402853270629040377454, −10.01995093726069598878330993587, −8.467368070231251108577606437796, −7.66216107986223151733029519108, −6.70496152593544578639994086916, −5.61153492403391391604303389884, −4.38847175936169729385174991285, −3.96084159815244289377413658045, −1.93973657953111178776505823991,
1.39277743682087190828840128906, 2.59524534362457127378917697732, 4.47061506942048281765745700127, 5.09800905159498476658462088282, 6.18557625203512561311730126408, 7.25293525731869865508733572228, 8.530392440369362893829674397850, 9.091157061229725669292686294597, 10.60056029780629329673421038406, 11.54665343075989333041231188851