L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.5 − 2.59i)5-s + (−2 − 1.73i)7-s − 0.999·8-s + (−1.5 − 2.59i)10-s − 4·13-s + (−2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s − 3·20-s + (3 − 5.19i)23-s + (−2 − 3.46i)25-s + (−2 + 3.46i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (−0.474 − 0.821i)10-s − 1.10·13-s + (−0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s − 0.670·20-s + (0.625 − 1.08i)23-s + (−0.400 − 0.692i)25-s + (−0.392 + 0.679i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.664470 - 1.34039i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.664470 - 1.34039i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.97521120396319300884388651646, −10.03214955822986258940058106115, −9.487501382713376966219188141374, −8.519028376233095419155086254444, −7.18554859597472449991759226715, −5.96820098628800440803179429790, −5.00860206016266275502081451712, −4.02971640661567738935830614593, −2.55527815226750860225589354216, −0.932461199028361631564426305531,
2.55751672897887108701148856704, 3.39487184518830954610970145201, 5.20903032460717302478741055474, 5.88852546313343139941398961503, 7.01591207653807893550005231314, 7.49696064333577431235420179954, 9.138280791539967093107697946641, 9.710471221659176654718370489850, 10.65230570450221215349824898259, 11.92358990998088304216342319944