| L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.669 + 0.743i)3-s + (−0.978 + 0.207i)4-s + (1.10 − 0.490i)5-s + (−0.809 − 0.587i)6-s + (0.798 − 2.52i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + (0.603 + 1.04i)10-s + (3.21 − 0.825i)11-s + (0.5 − 0.866i)12-s + (3.40 − 2.47i)13-s + (2.59 + 0.530i)14-s + (−0.372 + 1.14i)15-s + (0.913 − 0.406i)16-s + (−0.453 + 4.31i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.386 + 0.429i)3-s + (−0.489 + 0.103i)4-s + (0.492 − 0.219i)5-s + (−0.330 − 0.239i)6-s + (0.301 − 0.953i)7-s + (−0.109 − 0.336i)8-s + (−0.0348 − 0.331i)9-s + (0.190 + 0.330i)10-s + (0.968 − 0.249i)11-s + (0.144 − 0.249i)12-s + (0.943 − 0.685i)13-s + (0.692 + 0.141i)14-s + (−0.0962 + 0.296i)15-s + (0.228 − 0.101i)16-s + (−0.109 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38613 + 0.438723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38613 + 0.438723i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 7 | \( 1 + (-0.798 + 2.52i)T \) |
| 11 | \( 1 + (-3.21 + 0.825i)T \) |
| good | 5 | \( 1 + (-1.10 + 0.490i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-3.40 + 2.47i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.453 - 4.31i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.0256 - 0.00544i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.68 + 4.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.918 - 2.82i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.45 - 1.09i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (4.33 + 4.81i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-0.792 - 2.44i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 + (-11.1 - 2.36i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-10.7 - 4.80i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-2.09 + 0.445i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (6.21 - 2.76i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (1.64 + 2.84i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.42 + 5.39i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.73 - 0.580i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (1.31 + 12.4i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (1.92 + 1.39i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.52 + 2.64i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.2 - 11.0i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.80041429887110769556983653411, −10.42646369095440472504136359934, −9.162180572946268718897888350172, −8.522109050516114956304397923187, −7.34780298369425169736260834398, −6.33436999085191915023011746208, −5.62680263681030425589968442029, −4.41230203995567566298326320922, −3.59763068116143221327147435761, −1.17156207448042369614077686857,
1.45101365955841017452035258981, 2.54873377047127923975964181397, 4.04477096878241361876643093711, 5.30872396841632657425799926272, 6.15987767919642234136033078689, 7.14093026281964168575583044237, 8.555398951864646796109041447179, 9.230839247372197726310936514278, 10.09600376454911211441154385858, 11.37955948630230497418851203287
