| L(s) = 1 | + (1.12 + 0.602i)3-s + (−1.29 + 1.58i)5-s + (1.93 + 1.29i)7-s + (−0.758 − 1.13i)9-s + (4.58 + 1.39i)11-s + (−1.77 + 1.45i)13-s + (−2.41 + 1.00i)15-s + (−0.698 − 0.289i)17-s + (−0.355 + 3.60i)19-s + (1.40 + 2.62i)21-s + (0.824 + 0.164i)23-s + (0.157 + 0.793i)25-s + (−0.547 − 5.55i)27-s + (2.64 + 8.70i)29-s + (−4.31 + 4.31i)31-s + ⋯ |
| L(s) = 1 | + (0.651 + 0.348i)3-s + (−0.580 + 0.707i)5-s + (0.732 + 0.489i)7-s + (−0.252 − 0.378i)9-s + (1.38 + 0.419i)11-s + (−0.491 + 0.403i)13-s + (−0.624 + 0.258i)15-s + (−0.169 − 0.0701i)17-s + (−0.0815 + 0.827i)19-s + (0.306 + 0.573i)21-s + (0.171 + 0.0342i)23-s + (0.0315 + 0.158i)25-s + (−0.105 − 1.06i)27-s + (0.490 + 1.61i)29-s + (−0.775 + 0.775i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38629 + 0.979226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38629 + 0.979226i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-1.12 - 0.602i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (1.29 - 1.58i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.93 - 1.29i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.58 - 1.39i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (1.77 - 1.45i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (0.698 + 0.289i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.355 - 3.60i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-0.824 - 0.164i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-2.64 - 8.70i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (4.31 - 4.31i)T - 31iT^{2} \) |
| 37 | \( 1 + (3.24 - 0.319i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-2.34 + 11.7i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-7.17 + 3.83i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-1.13 + 2.74i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.85 + 9.39i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-3.31 - 2.71i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-0.0672 + 0.125i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (3.12 - 5.84i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-6.30 + 9.44i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (4.87 - 3.25i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (4.87 + 11.7i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-13.3 - 1.31i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (13.5 - 2.69i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-4.07 + 4.07i)T - 97iT^{2} \) |
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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
−11.11931556702195764537727131643, −10.16343871290709239234143647526, −8.983593058397287061474856181643, −8.735350142371235055231210390738, −7.38484442816146747257733150779, −6.73391059160185035438786267883, −5.37357345798896091749286024924, −4.07674201173503803891498988618, −3.33837159674419767469494248892, −1.90703477693042742214249813568,
1.04073406804362147489643036419, 2.56996338069764609959041765281, 4.04270169353643668219660051284, 4.78179186459003832089285297476, 6.15676099259777854702789481930, 7.42306367367837496234870830181, 8.039078423889763538915662361336, 8.774582630585010400534776768614, 9.591419068941279849495435292179, 10.99621991254237994889620392875
