| L(s) = 1 | + (−0.472 − 1.18i)2-s + (−1.42 + 0.977i)3-s + (0.266 − 0.252i)4-s + (−0.187 + 0.158i)5-s + (1.83 + 1.23i)6-s + (2.43 − 1.46i)7-s + (−2.74 − 1.27i)8-s + (1.08 − 2.79i)9-s + (0.277 + 0.146i)10-s + (−0.109 − 2.02i)11-s + (−0.134 + 0.622i)12-s + (0.464 − 2.10i)13-s + (−2.88 − 2.19i)14-s + (0.112 − 0.409i)15-s + (−0.169 + 3.12i)16-s + (1.22 − 2.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.334 − 0.839i)2-s + (−0.825 + 0.564i)3-s + (0.133 − 0.126i)4-s + (−0.0836 + 0.0710i)5-s + (0.749 + 0.504i)6-s + (0.918 − 0.552i)7-s + (−0.970 − 0.449i)8-s + (0.363 − 0.931i)9-s + (0.0875 + 0.0464i)10-s + (−0.0330 − 0.610i)11-s + (−0.0388 + 0.179i)12-s + (0.128 − 0.585i)13-s + (−0.771 − 0.586i)14-s + (0.0289 − 0.105i)15-s + (−0.0423 + 0.781i)16-s + (0.297 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.532045 - 0.631557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.532045 - 0.631557i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.42 - 0.977i)T \) |
| 59 | \( 1 + (-5.00 - 5.82i)T \) |
| good | 2 | \( 1 + (0.472 + 1.18i)T + (-1.45 + 1.37i)T^{2} \) |
| 5 | \( 1 + (0.187 - 0.158i)T + (0.808 - 4.93i)T^{2} \) |
| 7 | \( 1 + (-2.43 + 1.46i)T + (3.27 - 6.18i)T^{2} \) |
| 11 | \( 1 + (0.109 + 2.02i)T + (-10.9 + 1.18i)T^{2} \) |
| 13 | \( 1 + (-0.464 + 2.10i)T + (-11.7 - 5.45i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 2.03i)T + (-7.96 - 15.0i)T^{2} \) |
| 19 | \( 1 + (1.36 + 4.90i)T + (-16.2 + 9.79i)T^{2} \) |
| 23 | \( 1 + (1.16 - 1.71i)T + (-8.51 - 21.3i)T^{2} \) |
| 29 | \( 1 + (-6.90 - 2.75i)T + (21.0 + 19.9i)T^{2} \) |
| 31 | \( 1 + (-1.87 - 0.520i)T + (26.5 + 15.9i)T^{2} \) |
| 37 | \( 1 + (-2.61 - 5.64i)T + (-23.9 + 28.1i)T^{2} \) |
| 41 | \( 1 + (-9.17 + 6.22i)T + (15.1 - 38.0i)T^{2} \) |
| 43 | \( 1 + (9.20 + 0.499i)T + (42.7 + 4.64i)T^{2} \) |
| 47 | \( 1 + (1.56 - 1.83i)T + (-7.60 - 46.3i)T^{2} \) |
| 53 | \( 1 + (4.43 - 2.35i)T + (29.7 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.658 + 0.262i)T + (44.2 - 41.9i)T^{2} \) |
| 67 | \( 1 + (3.53 - 7.63i)T + (-43.3 - 51.0i)T^{2} \) |
| 71 | \( 1 + (-2.50 - 2.13i)T + (11.4 + 70.0i)T^{2} \) |
| 73 | \( 1 + (-1.17 + 1.54i)T + (-19.5 - 70.3i)T^{2} \) |
| 79 | \( 1 + (8.46 + 0.920i)T + (77.1 + 16.9i)T^{2} \) |
| 83 | \( 1 + (8.51 + 2.86i)T + (66.0 + 50.2i)T^{2} \) |
| 89 | \( 1 + (-4.91 + 12.3i)T + (-64.6 - 61.2i)T^{2} \) |
| 97 | \( 1 + (-5.55 - 7.30i)T + (-25.9 + 93.4i)T^{2} \) |
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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
−11.89419445523482648541549991996, −11.26639257113891945871194136131, −10.66749857519089273906354501172, −9.818613971955860621855159452327, −8.655598291366432508936736731919, −7.13161923401116932915472986123, −5.87123451535938561515910259409, −4.69385476713639692274573797608, −3.16480883730364073766473434283, −1.00446456473343103511120657629,
2.07448596546713081765169246774, 4.57452421383301936730126410981, 5.87945165654390999059784416718, 6.63288527444596822171181883547, 7.910019053472950058239206392463, 8.341263222203287918674386013596, 9.968032310162438510914003447798, 11.25454990501283806956461608946, 12.02608143232995636629370412604, 12.61175642915923845842296460869
