| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.389 − 1.68i)3-s + (0.173 − 0.984i)4-s + (−0.0926 + 0.525i)5-s + (−1.38 − 1.04i)6-s + (−2.63 + 0.214i)7-s + (−0.500 − 0.866i)8-s + (−2.69 + 1.31i)9-s + (0.266 + 0.462i)10-s + (−0.646 − 3.66i)11-s + (−1.72 + 0.0901i)12-s + (1.06 − 6.06i)13-s + (−1.88 + 1.85i)14-s + (0.923 − 0.0481i)15-s + (−0.939 − 0.342i)16-s + (−1.09 − 1.89i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.224 − 0.974i)3-s + (0.0868 − 0.492i)4-s + (−0.0414 + 0.235i)5-s + (−0.564 − 0.425i)6-s + (−0.996 + 0.0811i)7-s + (−0.176 − 0.306i)8-s + (−0.899 + 0.437i)9-s + (0.0844 + 0.146i)10-s + (−0.194 − 1.10i)11-s + (−0.499 + 0.0260i)12-s + (0.296 − 1.68i)13-s + (−0.502 + 0.496i)14-s + (0.238 − 0.0124i)15-s + (−0.234 − 0.0855i)16-s + (−0.264 − 0.458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.247400 - 1.19961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247400 - 1.19961i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.389 + 1.68i)T \) |
| 7 | \( 1 + (2.63 - 0.214i)T \) |
| good | 5 | \( 1 + (0.0926 - 0.525i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.646 + 3.66i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.06 + 6.06i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.09 + 1.89i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.35 - 2.35i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.36 + 1.14i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.83 - 10.4i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.267 + 1.51i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + (-1.45 + 8.27i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.99 + 4.19i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (1.81 + 10.3i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (3.95 - 6.84i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.66 - 0.969i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.02 + 5.82i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.89 + 2.42i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (4.58 - 7.93i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.60T + 73T^{2} \) |
| 79 | \( 1 + (-11.3 + 9.51i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.348 - 1.97i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (1.91 - 3.32i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.72 - 4.80i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.85290998227466349172631384744, −10.58943997127899480646701978721, −9.089304322676572482622485879301, −8.100337789828688858863992923882, −6.96277310112372062314194304887, −6.03089863204790024913467696249, −5.36458208181186723962680368323, −3.42721338367417524635774990150, −2.69581216101472264924167967808, −0.70038267110684288767271912502,
2.67342014903561780239312176684, 4.24967536381267925080301782766, 4.51703977254050118909315475610, 6.12038608567871396014782286387, 6.64055080735990844738179951082, 8.036892533587616669734543566282, 9.316428602297831622103013436936, 9.658531128875260830894206332799, 10.92931100184219928072100817549, 11.78552080031308264133434008678
