L(s) = 1 | + (0.187 − 0.478i)2-s + (0.742 − 1.56i)3-s + (1.27 + 1.18i)4-s + (−0.268 + 0.129i)5-s + (−0.609 − 0.648i)6-s + (−0.691 + 2.55i)7-s + (1.72 − 0.833i)8-s + (−1.89 − 2.32i)9-s + (0.0114 + 0.152i)10-s + (2.99 + 3.75i)11-s + (2.79 − 1.11i)12-s + (2.29 − 5.84i)13-s + (1.09 + 0.810i)14-s + (0.00304 + 0.517i)15-s + (0.185 + 2.47i)16-s + (5.90 − 5.47i)17-s + ⋯ |
L(s) = 1 | + (0.132 − 0.338i)2-s + (0.428 − 0.903i)3-s + (0.636 + 0.590i)4-s + (−0.120 + 0.0579i)5-s + (−0.248 − 0.264i)6-s + (−0.261 + 0.965i)7-s + (0.611 − 0.294i)8-s + (−0.632 − 0.774i)9-s + (0.00362 + 0.0483i)10-s + (0.903 + 1.13i)11-s + (0.806 − 0.321i)12-s + (0.636 − 1.62i)13-s + (0.291 + 0.216i)14-s + (0.000786 + 0.133i)15-s + (0.0464 + 0.619i)16-s + (1.43 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.90929 - 0.630663i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.90929 - 0.630663i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.742 + 1.56i)T \) |
| 7 | \( 1 + (0.691 - 2.55i)T \) |
good | 2 | \( 1 + (-0.187 + 0.478i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (0.268 - 0.129i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.99 - 3.75i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.29 + 5.84i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-5.90 + 5.47i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.00 - 3.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.15 + 5.05i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (5.43 + 1.67i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (2.70 - 4.68i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.43 + 0.442i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-0.475 - 0.324i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (3.57 - 2.43i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (0.940 - 2.39i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (0.647 - 0.199i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (3.50 - 2.39i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-7.54 + 6.99i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (1.93 - 3.35i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.893 + 3.91i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (11.0 + 1.66i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-1.33 - 2.31i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.06 + 5.26i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-1.74 - 4.45i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (3.88 - 6.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.39157154785966899869002003186, −10.13904797594653374176417134296, −9.142799786726681940373055065243, −8.074185376516236165674134642818, −7.46188664167119124719480576556, −6.50037510096082192913255913372, −5.48094713202976903138712055040, −3.60319682136442736523821450600, −2.83656534374524411544837217193, −1.59575856150501417034249327823,
1.61072674414181175449058335197, 3.54945771096862905430089273612, 4.15900560838930214199796963194, 5.63156276307679862991392578786, 6.41838654306114900247981937946, 7.48382672741924847902039225050, 8.572108946293985800382133655216, 9.488684601060973804158479264066, 10.32069883151148264070470415907, 11.18748009797250449107209181255