| L(s) = 1 | + (0.673 − 0.565i)2-s + (1.87 + 0.684i)3-s + (−0.213 + 1.20i)4-s + (−0.286 − 1.62i)5-s + (1.65 − 0.601i)6-s + (0.0603 − 0.104i)7-s + (1.41 + 2.45i)8-s + (0.766 + 0.642i)9-s + (−1.11 − 0.934i)10-s + (−0.5 − 0.866i)11-s + (−1.22 + 2.12i)12-s + (4.29 − 1.56i)13-s + (−0.0184 − 0.104i)14-s + (0.573 − 3.25i)15-s + (0.0393 + 0.0143i)16-s + (−5.26 + 4.42i)17-s + ⋯ |
| L(s) = 1 | + (0.476 − 0.399i)2-s + (1.08 + 0.394i)3-s + (−0.106 + 0.604i)4-s + (−0.128 − 0.727i)5-s + (0.674 − 0.245i)6-s + (0.0227 − 0.0394i)7-s + (0.501 + 0.868i)8-s + (0.255 + 0.214i)9-s + (−0.352 − 0.295i)10-s + (−0.150 − 0.261i)11-s + (−0.354 + 0.613i)12-s + (1.19 − 0.433i)13-s + (−0.00492 − 0.0279i)14-s + (0.148 − 0.840i)15-s + (0.00984 + 0.00358i)16-s + (−1.27 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.000652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93890 - 0.000632763i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93890 - 0.000632763i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (4.11 + 1.43i)T \) |
| good | 2 | \( 1 + (-0.673 + 0.565i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (-1.87 - 0.684i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.286 + 1.62i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.0603 + 0.104i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-4.29 + 1.56i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (5.26 - 4.42i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.726 - 4.12i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.94 + 4.98i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.08 + 5.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + (2.09 + 0.761i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.75 + 9.95i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.54 - 4.65i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.11 - 12.0i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-3.17 + 2.66i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.54 - 8.78i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.63 - 3.04i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.0748 + 0.424i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.36 - 1.95i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.77 - 1.01i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.54 - 2.67i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.90 + 3.60i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-4.25 + 3.57i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.67109142004801675048858922325, −11.46649528996944611130023696892, −10.59762062026394389855375098034, −8.981704663711954175725009024895, −8.624652526637896777934071020209, −7.76888662950196332720724127822, −5.94971436822779247623289480951, −4.31837441935652600246917496099, −3.71663853604243917382458143156, −2.31583992224896671881796508911,
2.05971847318127599786915505869, 3.54493958368815681251807289848, 4.90307114254482023278700128650, 6.47756181148488834194127251103, 7.02809150292351022042164292422, 8.416991141444013332468348050477, 9.201198403649404301908345640743, 10.51660109599588438369678682999, 11.24828321907232106839711043765, 12.89670450424714304257160458531
