| 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
−10.99621991254237994889620392875, −9.591419068941279849495435292179, −8.774582630585010400534776768614, −8.039078423889763538915662361336, −7.42306367367837496234870830181, −6.15676099259777854702789481930, −4.78179186459003832089285297476, −4.04270169353643668219660051284, −2.56996338069764609959041765281, −1.04073406804362147489643036419,
1.90703477693042742214249813568, 3.33837159674419767469494248892, 4.07674201173503803891498988618, 5.37357345798896091749286024924, 6.73391059160185035438786267883, 7.38484442816146747257733150779, 8.735350142371235055231210390738, 8.983593058397287061474856181643, 10.16343871290709239234143647526, 11.11931556702195764537727131643
