| L(s) = 1 | + (0.283 + 0.711i)2-s + (1.61 + 0.618i)3-s + (1.02 − 0.971i)4-s + (−1.06 + 0.905i)5-s + (0.0188 + 1.32i)6-s + (−2.13 + 1.28i)7-s + (2.37 + 1.09i)8-s + (2.23 + 2.00i)9-s + (−0.946 − 0.501i)10-s + (−0.330 − 6.10i)11-s + (2.26 − 0.938i)12-s + (−0.904 + 4.11i)13-s + (−1.51 − 1.15i)14-s + (−2.28 + 0.806i)15-s + (0.0445 − 0.822i)16-s + (0.896 − 1.48i)17-s + ⋯ |
| L(s) = 1 | + (0.200 + 0.503i)2-s + (0.934 + 0.356i)3-s + (0.512 − 0.485i)4-s + (−0.476 + 0.405i)5-s + (0.00768 + 0.541i)6-s + (−0.805 + 0.484i)7-s + (0.838 + 0.388i)8-s + (0.745 + 0.666i)9-s + (−0.299 − 0.158i)10-s + (−0.0997 − 1.83i)11-s + (0.652 − 0.270i)12-s + (−0.250 + 1.14i)13-s + (−0.405 − 0.308i)14-s + (−0.590 + 0.208i)15-s + (0.0111 − 0.205i)16-s + (0.217 − 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53365 + 0.660892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53365 + 0.660892i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.61 - 0.618i)T \) |
| 59 | \( 1 + (-6.80 - 3.56i)T \) |
| good | 2 | \( 1 + (-0.283 - 0.711i)T + (-1.45 + 1.37i)T^{2} \) |
| 5 | \( 1 + (1.06 - 0.905i)T + (0.808 - 4.93i)T^{2} \) |
| 7 | \( 1 + (2.13 - 1.28i)T + (3.27 - 6.18i)T^{2} \) |
| 11 | \( 1 + (0.330 + 6.10i)T + (-10.9 + 1.18i)T^{2} \) |
| 13 | \( 1 + (0.904 - 4.11i)T + (-11.7 - 5.45i)T^{2} \) |
| 17 | \( 1 + (-0.896 + 1.48i)T + (-7.96 - 15.0i)T^{2} \) |
| 19 | \( 1 + (1.94 + 7.00i)T + (-16.2 + 9.79i)T^{2} \) |
| 23 | \( 1 + (2.65 - 3.92i)T + (-8.51 - 21.3i)T^{2} \) |
| 29 | \( 1 + (7.17 + 2.85i)T + (21.0 + 19.9i)T^{2} \) |
| 31 | \( 1 + (-2.76 - 0.768i)T + (26.5 + 15.9i)T^{2} \) |
| 37 | \( 1 + (1.90 + 4.11i)T + (-23.9 + 28.1i)T^{2} \) |
| 41 | \( 1 + (-3.35 + 2.27i)T + (15.1 - 38.0i)T^{2} \) |
| 43 | \( 1 + (5.23 + 0.283i)T + (42.7 + 4.64i)T^{2} \) |
| 47 | \( 1 + (-4.44 + 5.22i)T + (-7.60 - 46.3i)T^{2} \) |
| 53 | \( 1 + (8.00 - 4.24i)T + (29.7 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-9.98 + 3.97i)T + (44.2 - 41.9i)T^{2} \) |
| 67 | \( 1 + (3.89 - 8.41i)T + (-43.3 - 51.0i)T^{2} \) |
| 71 | \( 1 + (-5.67 - 4.82i)T + (11.4 + 70.0i)T^{2} \) |
| 73 | \( 1 + (0.0289 - 0.0381i)T + (-19.5 - 70.3i)T^{2} \) |
| 79 | \( 1 + (-1.41 - 0.153i)T + (77.1 + 16.9i)T^{2} \) |
| 83 | \( 1 + (-5.76 - 1.94i)T + (66.0 + 50.2i)T^{2} \) |
| 89 | \( 1 + (4.09 - 10.2i)T + (-64.6 - 61.2i)T^{2} \) |
| 97 | \( 1 + (-3.43 - 4.51i)T + (-25.9 + 93.4i)T^{2} \) |
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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
−13.29440800086349382165556489207, −11.55048155091984380208229158985, −10.96205042781805081171187392788, −9.626608224248968484117772865686, −8.808599984910486613347074035170, −7.54345474340651344689324772943, −6.62440933037020310666640200815, −5.41734796759865063864904770749, −3.75245377662215546173027834595, −2.51761702188002493086550506367,
1.98424049589787997258637884914, 3.41209032513891690785419855103, 4.33622493165631136691487103669, 6.56622075892362515153081130966, 7.62361426648233753720408155745, 8.163357305110497670049290036044, 9.878001808447484331662261824851, 10.31017986754985892964376278715, 12.08966508586271018527287522784, 12.67283424958281265133204683920
