| L(s) = 1 | + (0.978 + 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (−1.28 − 1.42i)5-s + (−0.309 + 0.951i)6-s + (2.40 − 1.10i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.958 − 1.66i)10-s + (3.31 − 0.0907i)11-s + (−0.5 + 0.866i)12-s + (1.60 + 4.92i)13-s + (2.58 − 0.583i)14-s + (1.55 − 1.12i)15-s + (0.669 + 0.743i)16-s + (6.64 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (−0.0603 + 0.574i)3-s + (0.456 + 0.203i)4-s + (−0.573 − 0.637i)5-s + (−0.126 + 0.388i)6-s + (0.908 − 0.418i)7-s + (0.286 + 0.207i)8-s + (−0.326 − 0.0693i)9-s + (−0.303 − 0.525i)10-s + (0.999 − 0.0273i)11-s + (−0.144 + 0.250i)12-s + (0.443 + 1.36i)13-s + (0.689 − 0.155i)14-s + (0.400 − 0.290i)15-s + (0.167 + 0.185i)16-s + (1.61 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09697 + 0.509652i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09697 + 0.509652i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.104 - 0.994i)T \) |
| 7 | \( 1 + (-2.40 + 1.10i)T \) |
| 11 | \( 1 + (-3.31 + 0.0907i)T \) |
| good | 5 | \( 1 + (1.28 + 1.42i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.60 - 4.92i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-6.64 + 1.41i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (6.07 - 2.70i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.85 + 4.95i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.63 - 2.64i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (6.32 - 7.02i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.419 + 3.98i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (1.53 + 1.11i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 0.609T + 43T^{2} \) |
| 47 | \( 1 + (-5.35 + 2.38i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (1.66 - 1.85i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (11.9 + 5.34i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (4.34 + 4.82i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.15 - 3.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.23 + 3.78i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (13.3 + 5.93i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-6.90 - 1.46i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (2.12 - 6.54i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (3.97 - 6.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.48 + 4.57i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.18271129499850790898426670143, −10.53195874283463315143933626223, −9.112024258276612688118114505162, −8.490040583188525464149048878039, −7.39279804988633376076710685131, −6.35172837736993712845656407159, −5.11692187154554196565115965032, −4.29361049960597301392314875405, −3.66473831306625612783788428156, −1.58600952403208114921033476779,
1.48376358718737175168409195748, 3.00261200700558016594430258729, 3.98964601336319679798309646079, 5.42091710429446360730864453331, 6.12867403712795548041250872235, 7.40427491838917006345311867839, 7.909586390700433321568092612358, 9.092844095242103490574541844079, 10.50414001536844004059470581211, 11.24522854056178535164507012907
