L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.42 − 0.822i)5-s + (2.58 − 0.551i)7-s − 0.999·8-s + (−1.42 + 0.822i)10-s + (−2.53 − 2.13i)11-s + 1.29i·13-s + (0.816 − 2.51i)14-s + (−0.5 + 0.866i)16-s + (−1.68 − 2.91i)17-s + (−0.340 − 0.196i)19-s + 1.64i·20-s + (−3.11 + 1.13i)22-s + (−0.455 − 0.262i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.636 − 0.367i)5-s + (0.978 − 0.208i)7-s − 0.353·8-s + (−0.450 + 0.259i)10-s + (−0.765 − 0.643i)11-s + 0.358i·13-s + (0.218 − 0.672i)14-s + (−0.125 + 0.216i)16-s + (−0.408 − 0.707i)17-s + (−0.0780 − 0.0450i)19-s + 0.367i·20-s + (−0.664 + 0.241i)22-s + (−0.0949 − 0.0548i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.085161336\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.085161336\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.58 + 0.551i)T \) |
| 11 | \( 1 + (2.53 + 2.13i)T \) |
good | 5 | \( 1 + (1.42 + 0.822i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 1.29iT - 13T^{2} \) |
| 17 | \( 1 + (1.68 + 2.91i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.340 + 0.196i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.455 + 0.262i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 31 | \( 1 + (2.86 + 4.97i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.50 + 2.61i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 10.5iT - 43T^{2} \) |
| 47 | \( 1 + (4.48 + 2.59i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.43 - 0.830i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.26 + 4.77i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.32 + 1.91i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.80 + 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.62iT - 71T^{2} \) |
| 73 | \( 1 + (8.82 - 5.09i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.22 - 4.17i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.98T + 83T^{2} \) |
| 89 | \( 1 + (8.32 + 4.80i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−9.188636393757497566279847101981, −8.286837819899681476054658431048, −7.82181740623109077467497065362, −6.69369800352468488204905235350, −5.54314900215426722760212093079, −4.76540184806355481147077132959, −4.10863455732502725025003138149, −2.96893533026761409123857389362, −1.81929921468656685374242857100, −0.37910373953835252573317218889,
1.85625972181798028394017615989, 3.14463339555017094736165989168, 4.17366513591653598085127359680, 5.00190002814603723365262112054, 5.71904481798262293930975913615, 6.90745635401665458153574261642, 7.48641106619316350519634391190, 8.257328604019824557506980493886, 8.789267095097162992186885807586, 10.10706507381657623763739843870