| L(s) = 1 | + (0.919 − 0.442i)2-s + (1.70 − 0.278i)3-s + (−0.597 + 0.749i)4-s + (3.10 − 0.958i)5-s + (1.44 − 1.01i)6-s + (−2.22 − 1.43i)7-s + (−0.671 + 2.94i)8-s + (2.84 − 0.950i)9-s + (2.43 − 2.25i)10-s + (0.141 − 1.88i)11-s + (−0.813 + 1.44i)12-s + (0.173 − 2.31i)13-s + (−2.67 − 0.339i)14-s + (5.04 − 2.50i)15-s + (0.259 + 1.13i)16-s + (−1.50 + 3.84i)17-s + ⋯ |
| L(s) = 1 | + (0.650 − 0.313i)2-s + (0.987 − 0.160i)3-s + (−0.298 + 0.374i)4-s + (1.38 − 0.428i)5-s + (0.591 − 0.413i)6-s + (−0.839 − 0.543i)7-s + (−0.237 + 1.04i)8-s + (0.948 − 0.316i)9-s + (0.769 − 0.713i)10-s + (0.0425 − 0.567i)11-s + (−0.234 + 0.417i)12-s + (0.0480 − 0.641i)13-s + (−0.715 − 0.0908i)14-s + (1.30 − 0.645i)15-s + (0.0647 + 0.283i)16-s + (−0.366 + 0.932i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.62366 - 0.659729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.62366 - 0.659729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 + 0.278i)T \) |
| 7 | \( 1 + (2.22 + 1.43i)T \) |
| good | 2 | \( 1 + (-0.919 + 0.442i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (-3.10 + 0.958i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.141 + 1.88i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-0.173 + 2.31i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (1.50 - 3.84i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (1.82 - 3.15i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.97 - 0.900i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (2.29 - 5.84i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + 9.28T + 31T^{2} \) |
| 37 | \( 1 + (-0.627 + 0.0946i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (3.50 + 3.25i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-2.01 + 1.86i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (2.77 - 1.33i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (7.33 + 1.10i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (2.32 + 10.1i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (-7.55 - 9.47i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 0.379T + 67T^{2} \) |
| 71 | \( 1 + (1.84 - 2.31i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.302 - 4.03i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + (1.29 + 17.2i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-4.50 + 3.07i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (2.02 + 3.51i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.91805099089274193521180957737, −10.07724409528423202273884716631, −9.085902525146319381735138595361, −8.638834555096329215433699999647, −7.40931564252839944154801557996, −6.19037161994373742633610410751, −5.21988621690162254162680310435, −3.83008219245243632372916328902, −3.09229409257515611674445316180, −1.75707324844178613839006522159,
2.03374420934892922271362784789, 3.07217683519181341848771199874, 4.45351354204459166877725673454, 5.44803559333107861920238201559, 6.54344060992829388102617734805, 7.07548794640077197871953007557, 8.925375819264621946256703912383, 9.470591208373041870251995740283, 9.826720345618470414577526161212, 10.99595125534984008517463672689
