L(s) = 1 | + (0.866 − 0.5i)2-s + (0.258 − 0.965i)3-s + (0.499 − 0.866i)4-s + (1.17 + 1.90i)5-s + (−0.258 − 0.965i)6-s + (1.97 − 3.41i)7-s − 0.999i·8-s + (−0.866 − 0.499i)9-s + (1.96 + 1.06i)10-s + (−1.48 + 5.56i)11-s + (−0.707 − 0.707i)12-s + (0.730 − 3.53i)13-s − 3.94i·14-s + (2.14 − 0.637i)15-s + (−0.5 − 0.866i)16-s + (3.85 − 1.03i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.149 − 0.557i)3-s + (0.249 − 0.433i)4-s + (0.523 + 0.851i)5-s + (−0.105 − 0.394i)6-s + (0.745 − 1.29i)7-s − 0.353i·8-s + (−0.288 − 0.166i)9-s + (0.621 + 0.336i)10-s + (−0.449 + 1.67i)11-s + (−0.204 − 0.204i)12-s + (0.202 − 0.979i)13-s − 1.05i·14-s + (0.553 − 0.164i)15-s + (−0.125 − 0.216i)16-s + (0.936 − 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.94344 - 1.08419i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94344 - 1.08419i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (-1.17 - 1.90i)T \) |
| 13 | \( 1 + (-0.730 + 3.53i)T \) |
good | 7 | \( 1 + (-1.97 + 3.41i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.48 - 5.56i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-3.85 + 1.03i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.08 - 0.558i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (4.18 + 1.12i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-5.04 + 2.91i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.83 - 2.83i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.82 + 3.16i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0577 + 0.0154i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-2.76 - 10.3i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 + (-4.15 - 4.15i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.509 - 1.89i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (1.42 - 2.47i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.35 - 1.36i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.31 - 12.3i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 4.59iT - 73T^{2} \) |
| 79 | \( 1 + 2.49iT - 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + (8.00 + 2.14i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-8.26 - 4.76i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.10733836671051279602875510864, −10.28010852051582735717995906940, −9.892955276190054842852738405785, −7.963955002578464848642968381062, −7.39130484225460368484362538959, −6.46936114706162841730930677246, −5.24684814975580750753244613980, −4.11157990133571249910755025196, −2.76714302399659932802651562808, −1.54242107241138421945686553428,
2.03719159209019725168882966550, 3.48701843591205211040111524355, 4.83628829609406952031614747578, 5.55235827169119234455021638056, 6.23594271116808041835373887181, 8.179344246988656383274327991868, 8.517098717709533836481278996798, 9.394871423616443413495804267126, 10.67753946158799259015789737740, 11.67956796682066051777785972183