| L(s) = 1 | + (−0.127 + 1.40i)2-s + (−1.99 − 0.346i)3-s + (−1.96 − 0.358i)4-s + (0.531 − 1.60i)5-s + (0.742 − 2.76i)6-s + (−0.215 + 4.37i)7-s + (0.754 − 2.72i)8-s + (1.04 + 0.374i)9-s + (2.19 + 0.953i)10-s + (−5.21 − 2.31i)11-s + (3.80 + 1.39i)12-s + (1.21 − 1.05i)13-s + (−6.14 − 0.859i)14-s + (−1.61 + 3.03i)15-s + (3.74 + 1.40i)16-s + (5.34 − 2.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.0899 + 0.995i)2-s + (−1.15 − 0.200i)3-s + (−0.983 − 0.179i)4-s + (0.237 − 0.719i)5-s + (0.303 − 1.13i)6-s + (−0.0813 + 1.65i)7-s + (0.266 − 0.963i)8-s + (0.348 + 0.124i)9-s + (0.695 + 0.301i)10-s + (−1.57 − 0.697i)11-s + (1.09 + 0.403i)12-s + (0.337 − 0.291i)13-s + (−1.64 − 0.229i)14-s + (−0.418 + 0.782i)15-s + (0.935 + 0.352i)16-s + (1.29 − 0.693i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.718781 + 0.101991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718781 + 0.101991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.127 - 1.40i)T \) |
| good | 3 | \( 1 + (1.99 + 0.346i)T + (2.82 + 1.01i)T^{2} \) |
| 5 | \( 1 + (-0.531 + 1.60i)T + (-4.01 - 2.97i)T^{2} \) |
| 7 | \( 1 + (0.215 - 4.37i)T + (-6.96 - 0.686i)T^{2} \) |
| 11 | \( 1 + (5.21 + 2.31i)T + (7.38 + 8.15i)T^{2} \) |
| 13 | \( 1 + (-1.21 + 1.05i)T + (1.90 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.34 + 2.85i)T + (9.44 - 14.1i)T^{2} \) |
| 19 | \( 1 + (-5.80 + 3.28i)T + (9.76 - 16.2i)T^{2} \) |
| 23 | \( 1 + (-0.734 - 2.93i)T + (-20.2 + 10.8i)T^{2} \) |
| 29 | \( 1 + (7.45 - 0.183i)T + (28.9 - 1.42i)T^{2} \) |
| 31 | \( 1 + (-3.23 + 2.15i)T + (11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-4.43 - 0.547i)T + (35.8 + 8.99i)T^{2} \) |
| 41 | \( 1 + (-6.90 + 5.12i)T + (11.9 - 39.2i)T^{2} \) |
| 43 | \( 1 + (-3.05 + 4.34i)T + (-14.4 - 40.4i)T^{2} \) |
| 47 | \( 1 + (0.332 + 3.37i)T + (-46.0 + 9.16i)T^{2} \) |
| 53 | \( 1 + (2.81 + 0.0691i)T + (52.9 + 2.60i)T^{2} \) |
| 59 | \( 1 + (-2.99 + 3.47i)T + (-8.65 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-9.07 + 2.03i)T + (55.1 - 26.0i)T^{2} \) |
| 67 | \( 1 + (-4.07 - 2.58i)T + (28.6 + 60.5i)T^{2} \) |
| 71 | \( 1 + (0.0135 + 0.00638i)T + (45.0 + 54.8i)T^{2} \) |
| 73 | \( 1 + (0.149 + 3.04i)T + (-72.6 + 7.15i)T^{2} \) |
| 79 | \( 1 + (-0.269 - 0.327i)T + (-15.4 + 77.4i)T^{2} \) |
| 83 | \( 1 + (16.4 - 2.02i)T + (80.5 - 20.1i)T^{2} \) |
| 89 | \( 1 + (-1.15 + 4.61i)T + (-78.4 - 41.9i)T^{2} \) |
| 97 | \( 1 + (0.0305 - 0.153i)T + (-89.6 - 37.1i)T^{2} \) |
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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
−11.07355875067246997193416828832, −9.766867666304958354696734024339, −9.076595763329541171572236972627, −8.176944502759057891221162374034, −7.28333056782455932267640943302, −5.82395981618675057918554615588, −5.43556876164447758954649907070, −5.20971009283381589870754930225, −3.01790764982010768883366859515, −0.68662399071995916968809171334,
1.04216781912978642779023844456, 2.88318408512270933238746086749, 4.05324239655914099432656808077, 5.06444528730433362036179715611, 6.00548201999461254742729752345, 7.37248363171942283922389122824, 7.990336473911824164519705790476, 9.921636825930125885733210415746, 10.09881004838767888757239428870, 10.85401966668871953553587155134
