| L(s) = 1 | + (0.311 + 0.647i)2-s + (1.16 − 2.41i)3-s + (2.17 − 2.72i)4-s + (−3.49 − 0.798i)5-s + 1.92·6-s + 9.27i·7-s + (5.24 + 1.19i)8-s + (1.13 + 1.42i)9-s + (−0.573 − 2.51i)10-s + (−7.85 − 9.85i)11-s + (−4.04 − 8.40i)12-s + (−5.05 + 22.1i)13-s + (−6.00 + 2.89i)14-s + (−5.99 + 7.51i)15-s + (−2.23 − 9.81i)16-s + (−4.36 − 19.1i)17-s + ⋯ |
| L(s) = 1 | + (0.155 + 0.323i)2-s + (0.387 − 0.804i)3-s + (0.542 − 0.680i)4-s + (−0.699 − 0.159i)5-s + 0.320·6-s + 1.32i·7-s + (0.655 + 0.149i)8-s + (0.126 + 0.158i)9-s + (−0.0573 − 0.251i)10-s + (−0.714 − 0.895i)11-s + (−0.337 − 0.700i)12-s + (−0.388 + 1.70i)13-s + (−0.429 + 0.206i)14-s + (−0.399 + 0.500i)15-s + (−0.139 − 0.613i)16-s + (−0.256 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.27744 - 0.232221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27744 - 0.232221i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-42.3 + 7.19i)T \) |
| good | 2 | \( 1 + (-0.311 - 0.647i)T + (-2.49 + 3.12i)T^{2} \) |
| 3 | \( 1 + (-1.16 + 2.41i)T + (-5.61 - 7.03i)T^{2} \) |
| 5 | \( 1 + (3.49 + 0.798i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 - 9.27iT - 49T^{2} \) |
| 11 | \( 1 + (7.85 + 9.85i)T + (-26.9 + 117. i)T^{2} \) |
| 13 | \( 1 + (5.05 - 22.1i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (4.36 + 19.1i)T + (-260. + 125. i)T^{2} \) |
| 19 | \( 1 + (-2.63 - 2.09i)T + (80.3 + 351. i)T^{2} \) |
| 23 | \( 1 + (-9.12 - 11.4i)T + (-117. + 515. i)T^{2} \) |
| 29 | \( 1 + (-0.220 - 0.457i)T + (-524. + 657. i)T^{2} \) |
| 31 | \( 1 + (6.91 - 3.33i)T + (599. - 751. i)T^{2} \) |
| 37 | \( 1 + 40.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (44.6 - 21.5i)T + (1.04e3 - 1.31e3i)T^{2} \) |
| 47 | \( 1 + (21.6 - 27.1i)T + (-491. - 2.15e3i)T^{2} \) |
| 53 | \( 1 + (16.3 + 71.4i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 + (-1.19 - 5.22i)T + (-3.13e3 + 1.51e3i)T^{2} \) |
| 61 | \( 1 + (-34.7 + 72.0i)T + (-2.32e3 - 2.90e3i)T^{2} \) |
| 67 | \( 1 + (37.1 - 46.5i)T + (-998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (4.50 + 3.58i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (-22.3 - 5.09i)T + (4.80e3 + 2.31e3i)T^{2} \) |
| 79 | \( 1 - 77.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-106. - 51.3i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (42.5 - 88.2i)T + (-4.93e3 - 6.19e3i)T^{2} \) |
| 97 | \( 1 + (13.0 + 16.3i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−15.82241753888633877525574808524, −14.49385226199406707999577551239, −13.54537764956520066662713931439, −12.05932977382954597698581322498, −11.23155019596640261479215342633, −9.289024700926593487593203915892, −7.902753200188013866600966089951, −6.71786936564995857216307261637, −5.16298958393678408042868388055, −2.27076397398909320008153392811,
3.29058324429967389988170370496, 4.39715636097415010145113301118, 7.18692477635485246606612510040, 8.074744322026725907338210590928, 10.20384570263979425630396287010, 10.71383088828765498188357638497, 12.36010780926015187761598293571, 13.25261531509087226348193820421, 15.08789335545477745713082936201, 15.51540943980675405768807586605
