L(s) = 1 | + (0.569 + 2.12i)2-s − i·3-s + (−2.46 + 1.42i)4-s + (0.837 − 3.12i)5-s + (2.12 − 0.569i)6-s + (0.780 − 2.52i)7-s + (−1.31 − 1.31i)8-s − 9-s + 7.12·10-s + (2.85 + 2.85i)11-s + (1.42 + 2.46i)12-s + (2.61 + 2.47i)13-s + (5.81 + 0.218i)14-s + (−3.12 − 0.837i)15-s + (−0.796 + 1.38i)16-s + (−3.80 − 6.59i)17-s + ⋯ |
L(s) = 1 | + (0.402 + 1.50i)2-s − 0.577i·3-s + (−1.23 + 0.711i)4-s + (0.374 − 1.39i)5-s + (0.868 − 0.232i)6-s + (0.294 − 0.955i)7-s + (−0.465 − 0.465i)8-s − 0.333·9-s + 2.25·10-s + (0.859 + 0.859i)11-s + (0.410 + 0.711i)12-s + (0.726 + 0.687i)13-s + (1.55 + 0.0584i)14-s + (−0.806 − 0.216i)15-s + (−0.199 + 0.345i)16-s + (−0.922 − 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58206 + 0.502170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58206 + 0.502170i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 + (-0.780 + 2.52i)T \) |
| 13 | \( 1 + (-2.61 - 2.47i)T \) |
good | 2 | \( 1 + (-0.569 - 2.12i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-0.837 + 3.12i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.85 - 2.85i)T + 11iT^{2} \) |
| 17 | \( 1 + (3.80 + 6.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.95 - 1.95i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.89 + 2.24i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.49 - 2.58i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.58 + 0.691i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (1.64 - 0.439i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (2.95 - 11.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.30 - 3.64i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.48 + 0.666i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.91 - 6.77i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.90 + 2.11i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 9.20iT - 61T^{2} \) |
| 67 | \( 1 + (-5.52 + 5.52i)T - 67iT^{2} \) |
| 71 | \( 1 + (0.721 + 2.69i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-0.453 - 1.69i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.31 - 7.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.63 + 1.63i)T + 83iT^{2} \) |
| 89 | \( 1 + (4.66 + 17.4i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (12.7 - 3.41i)T + (84.0 - 48.5i)T^{2} \) |
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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
−12.32312241412088168145114548915, −11.33405971600472215527899348126, −9.614231828983135025037697354170, −8.815410199084299035111539757423, −7.88504893936149912300852350155, −6.96056129598381914714124891252, −6.20336981184030573727950018179, −4.83647669560884578614793870815, −4.33722005252205877186705653922, −1.43513930230917192781230554531,
2.02539073775853282867438701364, 3.16895383082278814560872063834, 3.95458717929682004347683026011, 5.60422299514776962173660378239, 6.47484739626348364815222822214, 8.360804839251968326181692232050, 9.325164062387988299191252774941, 10.36999699699015567949446502633, 10.94203814620548455833670326446, 11.50335268868289187366308706635