| L(s) = 1 | + (1.27 − 0.614i)2-s + (0.941 + 0.336i)3-s + (1.24 − 1.56i)4-s + (−1.14 − 0.845i)5-s + (1.40 − 0.148i)6-s + (1.64 + 0.162i)7-s + (0.626 − 2.75i)8-s + (0.773 + 0.634i)9-s + (−1.97 − 0.377i)10-s + (2.57 + 2.84i)11-s + (1.70 − 1.05i)12-s + (0.517 − 3.48i)13-s + (2.20 − 0.806i)14-s + (−0.788 − 1.18i)15-s + (−0.895 − 3.89i)16-s + (1.72 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (0.900 − 0.434i)2-s + (0.543 + 0.194i)3-s + (0.622 − 0.782i)4-s + (−0.510 − 0.378i)5-s + (0.574 − 0.0608i)6-s + (0.623 + 0.0614i)7-s + (0.221 − 0.975i)8-s + (0.257 + 0.211i)9-s + (−0.623 − 0.119i)10-s + (0.776 + 0.856i)11-s + (0.490 − 0.304i)12-s + (0.143 − 0.967i)13-s + (0.588 − 0.215i)14-s + (−0.203 − 0.304i)15-s + (−0.223 − 0.974i)16-s + (0.418 − 0.626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.80585 - 1.43830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.80585 - 1.43830i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.27 + 0.614i)T \) |
| 3 | \( 1 + (-0.941 - 0.336i)T \) |
| good | 5 | \( 1 + (1.14 + 0.845i)T + (1.45 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-1.64 - 0.162i)T + (6.86 + 1.36i)T^{2} \) |
| 11 | \( 1 + (-2.57 - 2.84i)T + (-1.07 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.517 + 3.48i)T + (-12.4 - 3.77i)T^{2} \) |
| 17 | \( 1 + (-1.72 + 2.58i)T + (-6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (3.52 - 5.87i)T + (-8.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (2.50 - 1.33i)T + (12.7 - 19.1i)T^{2} \) |
| 29 | \( 1 + (-0.172 + 0.00848i)T + (28.8 - 2.84i)T^{2} \) |
| 31 | \( 1 + (-2.59 + 6.27i)T + (-21.9 - 21.9i)T^{2} \) |
| 37 | \( 1 + (-9.91 - 2.48i)T + (32.6 + 17.4i)T^{2} \) |
| 41 | \( 1 + (2.76 - 9.12i)T + (-34.0 - 22.7i)T^{2} \) |
| 43 | \( 1 + (2.35 + 6.58i)T + (-33.2 + 27.2i)T^{2} \) |
| 47 | \( 1 + (4.43 - 0.881i)T + (43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (7.64 + 0.375i)T + (52.7 + 5.19i)T^{2} \) |
| 59 | \( 1 + (-1.79 - 12.0i)T + (-56.4 + 17.1i)T^{2} \) |
| 61 | \( 1 + (-2.75 + 1.30i)T + (38.6 - 47.1i)T^{2} \) |
| 67 | \( 1 + (-6.10 - 12.9i)T + (-42.5 + 51.7i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 6.52i)T + (-13.8 + 69.6i)T^{2} \) |
| 73 | \( 1 + (2.36 - 0.232i)T + (71.5 - 14.2i)T^{2} \) |
| 79 | \( 1 + (2.31 - 11.6i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-2.17 + 0.545i)T + (73.1 - 39.1i)T^{2} \) |
| 89 | \( 1 + (0.883 + 0.472i)T + (49.4 + 74.0i)T^{2} \) |
| 97 | \( 1 + (12.3 + 5.11i)T + (68.5 + 68.5i)T^{2} \) |
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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
−10.04989185939550203180977953992, −9.742538243798061119683732272785, −8.260356378451056870976910319510, −7.79407224897219029539467769236, −6.53954146204926305033927779528, −5.50634112711255412862867253210, −4.43265448296968190423495178105, −3.91970193505022940092551753205, −2.62794517815119092472312578717, −1.38191951655660824357571499044,
1.83789348819505200650485743902, 3.18023127218846489128561759381, 4.00619720378317060129128645295, 4.87522433072379400233460947106, 6.29945194788718129745151723985, 6.77411107154480061671046096015, 7.86167068401137600626190125192, 8.440214743573065377480361342070, 9.332253086860566081326937483603, 10.93521312218269255831119246628
