| L(s) = 1 | + (−0.766 − 0.642i)2-s + (−1.73 − 0.0419i)3-s + (0.173 + 0.984i)4-s + (−0.354 − 0.128i)5-s + (1.29 + 1.14i)6-s + (0.173 − 0.984i)7-s + (0.500 − 0.866i)8-s + (2.99 + 0.145i)9-s + (0.188 + 0.326i)10-s + (2.76 − 1.00i)11-s + (−0.259 − 1.71i)12-s + (−4.07 + 3.42i)13-s + (−0.766 + 0.642i)14-s + (0.608 + 0.238i)15-s + (−0.939 + 0.342i)16-s + (−3.02 − 5.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.999 − 0.0241i)3-s + (0.0868 + 0.492i)4-s + (−0.158 − 0.0576i)5-s + (0.530 + 0.467i)6-s + (0.0656 − 0.372i)7-s + (0.176 − 0.306i)8-s + (0.998 + 0.0483i)9-s + (0.0596 + 0.103i)10-s + (0.832 − 0.303i)11-s + (−0.0748 − 0.494i)12-s + (−1.13 + 0.949i)13-s + (−0.204 + 0.171i)14-s + (0.156 + 0.0614i)15-s + (−0.234 + 0.0855i)16-s + (−0.734 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.113892 - 0.390652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113892 - 0.390652i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (1.73 + 0.0419i)T \) |
| 7 | \( 1 + (-0.173 + 0.984i)T \) |
| good | 5 | \( 1 + (0.354 + 0.128i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-2.76 + 1.00i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.07 - 3.42i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.02 + 5.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0148 + 0.0256i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.37 + 7.81i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.78 + 4.01i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.0693 + 0.393i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.75 - 4.77i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.65 - 4.74i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.46 - 0.531i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.89 + 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 4.70T + 53T^{2} \) |
| 59 | \( 1 + (-6.22 - 2.26i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.33 + 13.2i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.08 + 7.61i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.81 - 4.87i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.51 - 9.55i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.32 + 2.79i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.38 - 7.03i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.32 + 2.28i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.37 - 3.04i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.23880976206829575049354671843, −10.00751458363818586599364975588, −9.502821290002570906342679323441, −8.250258228366100261260579618101, −7.03892055497106875535110667065, −6.49116624700842553566641773851, −4.86880034639831647028728542883, −4.06710492087380634572426944404, −2.15514647926568377148826382135, −0.36448191758729469063586101494,
1.71705050509711810793570997078, 3.90767189929148712599697483744, 5.25360315045918001520389789811, 5.96488086543851812581999998869, 7.08524414955153176464895839838, 7.78890717656917405926453929472, 9.133156604483244160115693043530, 9.875766699308422988218647111143, 10.80626523899257446734915998335, 11.60666794070204818929409545866
