| L(s) = 1 | + (−0.750 − 0.159i)2-s + (−1.69 − 0.338i)3-s + (−1.28 − 0.573i)4-s + (−0.934 − 2.03i)5-s + (1.22 + 0.525i)6-s + (−1.54 + 2.66i)7-s + (2.11 + 1.53i)8-s + (2.77 + 1.15i)9-s + (0.377 + 1.67i)10-s + (2.10 + 0.447i)11-s + (1.99 + 1.41i)12-s + (−1.99 + 0.423i)13-s + (1.58 − 1.75i)14-s + (0.899 + 3.76i)15-s + (0.542 + 0.602i)16-s + (6.22 + 4.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.531 − 0.112i)2-s + (−0.980 − 0.195i)3-s + (−0.644 − 0.286i)4-s + (−0.417 − 0.908i)5-s + (0.498 + 0.214i)6-s + (−0.582 + 1.00i)7-s + (0.748 + 0.544i)8-s + (0.923 + 0.383i)9-s + (0.119 + 0.529i)10-s + (0.634 + 0.134i)11-s + (0.575 + 0.407i)12-s + (−0.552 + 0.117i)13-s + (0.423 − 0.469i)14-s + (0.232 + 0.972i)15-s + (0.135 + 0.150i)16-s + (1.51 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.281615 + 0.191595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.281615 + 0.191595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.69 + 0.338i)T \) |
| 5 | \( 1 + (0.934 + 2.03i)T \) |
| good | 2 | \( 1 + (0.750 + 0.159i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (1.54 - 2.66i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.10 - 0.447i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.99 - 0.423i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-6.22 - 4.52i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.68 + 1.95i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.89 - 4.32i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.382 + 3.64i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.995 - 9.47i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (1.42 - 4.40i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-8.46 + 1.79i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.735 + 1.27i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.121 + 1.15i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (5.64 - 4.10i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.193 - 0.0410i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (0.235 + 0.0499i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (1.30 - 12.4i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (5.29 - 3.84i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.49 + 7.67i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.18 - 11.2i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (5.53 - 2.46i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.671 + 2.06i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.64 + 15.6i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.37474106646968478007263770833, −11.64536384427469911893897946530, −10.31679603046948861071177943426, −9.541970274937644812725417382980, −8.659252127556838510410477541448, −7.60998425897414735526169893428, −6.03441983954576931755224890983, −5.26240900167962067308156300089, −4.09605435451686936223621728983, −1.43455387873529942466659545713,
0.42650937331544843851679065490, 3.56779014218296693277460121617, 4.43947323940957521559289285989, 6.06958906050129408944413985979, 7.18139256721193155389965358641, 7.77301656252623392682253621158, 9.524709708136713486526341763184, 10.07442983701053786986190286736, 10.88848106318038069367244687194, 12.01177607189248909464155397589
