L(s) = 1 | − 0.593·5-s + (0.0665 + 2.64i)7-s − 0.593·11-s + (−1.25 − 2.17i)13-s + (−1.46 − 2.52i)17-s + (−2.69 + 4.66i)19-s + 4.46·23-s − 4.64·25-s + (3.09 − 5.36i)29-s + (−3.93 + 6.81i)31-s + (−0.0394 − 1.56i)35-s + (0.5 − 0.866i)37-s + (0.136 + 0.236i)41-s + (5.58 − 9.66i)43-s + (−6.08 − 10.5i)47-s + ⋯ |
L(s) = 1 | − 0.265·5-s + (0.0251 + 0.999i)7-s − 0.178·11-s + (−0.348 − 0.603i)13-s + (−0.354 − 0.613i)17-s + (−0.617 + 1.06i)19-s + 0.930·23-s − 0.929·25-s + (0.575 − 0.996i)29-s + (−0.706 + 1.22i)31-s + (−0.00667 − 0.265i)35-s + (0.0821 − 0.142i)37-s + (0.0213 + 0.0369i)41-s + (0.851 − 1.47i)43-s + (−0.887 − 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4368726735\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4368726735\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.0665 - 2.64i)T \) |
good | 5 | \( 1 + 0.593T + 5T^{2} \) |
| 11 | \( 1 + 0.593T + 11T^{2} \) |
| 13 | \( 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.46 + 2.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 + (-3.09 + 5.36i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.93 - 6.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.136 - 0.236i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.58 + 9.66i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.02 + 6.97i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.32 - 7.48i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.32 - 5.75i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.956 - 1.65i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + (-3.95 - 6.85i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.62 + 8.00i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.85 + 6.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.21 + 10.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.86 + 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.578521162049119006075566360218, −7.73318392628776622297481318340, −7.03702935186162363603925086759, −6.04614469582313632061930801385, −5.43937107592793750848811380115, −4.65314004553147250613803840458, −3.60193562419283687915475595589, −2.71364309057279877397518075675, −1.81387882155805111513604093031, −0.13811237703275275945094273193,
1.24972310711809231055193318103, 2.45532829379490195336438340564, 3.50563631466595771172317581845, 4.41357105382514782688308722520, 4.85402549008025551271074926955, 6.16758505230805104151677337661, 6.72810743527269302460310904144, 7.56834581849613335742772788736, 8.046921770047462294595826769028, 9.153083695996690936501550947041