| L(s) = 1 | + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.614 − 1.34i)5-s + (−0.142 + 0.989i)6-s + (3.07 − 0.902i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (1.41 + 0.416i)10-s + (−0.0362 − 0.0417i)11-s + (−0.654 − 0.755i)12-s + (0.773 + 0.227i)13-s + (−1.33 + 2.91i)14-s + (−1.24 − 0.799i)15-s + (−0.959 + 0.281i)16-s + (−0.293 + 2.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.274 − 0.601i)5-s + (−0.0580 + 0.404i)6-s + (1.16 − 0.341i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (0.448 + 0.131i)10-s + (−0.0109 − 0.0125i)11-s + (−0.189 − 0.218i)12-s + (0.214 + 0.0629i)13-s + (−0.355 + 0.778i)14-s + (−0.321 − 0.206i)15-s + (−0.239 + 0.0704i)16-s + (−0.0710 + 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04652 - 0.0698148i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04652 - 0.0698148i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (4.62 + 1.25i)T \) |
| good | 5 | \( 1 + (0.614 + 1.34i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-3.07 + 0.902i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.0362 + 0.0417i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.773 - 0.227i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.293 - 2.03i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.523 - 3.63i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (1.04 - 7.27i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.69 + 4.30i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.28 - 2.80i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (0.606 + 1.32i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (10.5 - 6.77i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + (-10.5 + 3.11i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-5.14 - 1.51i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-10.2 - 6.59i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (2.96 - 3.41i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-9.56 + 11.0i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.40 - 9.76i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (3.47 + 1.01i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (4.16 - 9.12i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (6.94 - 4.46i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.91 - 6.38i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−13.30810265200935688499897691017, −12.19836169487534474534790112997, −11.06300304521986880238619126664, −9.909307919035344275266719745690, −8.502895287859484182023023195514, −8.154861840195724372258267237208, −6.96037492367076562713978295900, −5.43351220196012783440834890659, −4.08340455478138784358097124592, −1.59472975524456984987053233662,
2.17932492208610412473641262205, 3.68011831078985214075761822149, 5.14818047363877719105536064508, 7.11040560983323582980650579585, 8.126686474302001767003250199076, 9.029575556089434822893765082251, 10.19581315810487905235192003331, 11.20753010065542825927028092387, 11.79332855353865899413342449667, 13.24720894213753036538423625282
