| L(s) = 1 | + (−0.978 + 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (1.51 − 1.67i)5-s + (−0.309 − 0.951i)6-s + (1.81 + 1.92i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (−1.13 + 1.95i)10-s + (1.24 − 3.07i)11-s + (0.5 + 0.866i)12-s + (−0.0900 + 0.276i)13-s + (−2.17 − 1.50i)14-s + (1.82 + 1.32i)15-s + (0.669 − 0.743i)16-s + (6.07 + 1.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 + 0.147i)2-s + (0.0603 + 0.574i)3-s + (0.456 − 0.203i)4-s + (0.676 − 0.751i)5-s + (−0.126 − 0.388i)6-s + (0.685 + 0.728i)7-s + (−0.286 + 0.207i)8-s + (−0.326 + 0.0693i)9-s + (−0.357 + 0.618i)10-s + (0.376 − 0.926i)11-s + (0.144 + 0.250i)12-s + (−0.0249 + 0.0768i)13-s + (−0.581 − 0.402i)14-s + (0.472 + 0.343i)15-s + (0.167 − 0.185i)16-s + (1.47 + 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27171 + 0.225923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27171 + 0.225923i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.81 - 1.92i)T \) |
| 11 | \( 1 + (-1.24 + 3.07i)T \) |
| good | 5 | \( 1 + (-1.51 + 1.67i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (0.0900 - 0.276i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-6.07 - 1.29i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (2.02 + 0.899i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (3.94 + 6.84i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.32 - 3.86i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.0833 - 0.0926i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.598 - 5.69i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-1.99 + 1.45i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + (-5.34 - 2.38i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-2.79 - 3.09i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (11.2 - 5.02i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (2.05 - 2.27i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (3.28 - 5.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.38 + 13.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.72 + 2.99i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (6.17 - 1.31i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (0.346 + 1.06i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (8.72 + 15.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.42 + 10.5i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.81662797476663527188938358773, −10.15978754179509317311519276942, −9.051326969205665306343476672610, −8.693704717186105154917005732590, −7.82968298713424751621755915543, −6.19576670751136235750650897864, −5.58082778995240985068054492421, −4.48647697217057985096194995421, −2.82262992926342995415982242218, −1.31957241053001368400998722166,
1.35819626638699658258163683638, 2.46663497210968984314792276080, 3.95079043472343751459947081448, 5.57129547549517988722647223439, 6.61366859398065643159814065342, 7.49870055062391276263405912325, 8.010910198483591921408923620997, 9.436189835355602493067399090782, 10.07872687525692530942813929676, 10.81809498211869900547169868655
