| 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
−11.24522854056178535164507012907, −10.50414001536844004059470581211, −9.092844095242103490574541844079, −7.909586390700433321568092612358, −7.40427491838917006345311867839, −6.12867403712795548041250872235, −5.42091710429446360730864453331, −3.98964601336319679798309646079, −3.00261200700558016594430258729, −1.48376358718737175168409195748,
1.58600952403208114921033476779, 3.66473831306625612783788428156, 4.29361049960597301392314875405, 5.11692187154554196565115965032, 6.35172837736993712845656407159, 7.39279804988633376076710685131, 8.490040583188525464149048878039, 9.112024258276612688118114505162, 10.53195874283463315143933626223, 11.18271129499850790898426670143
