L(s) = 1 | + (1 + 1.73i)3-s + (0.5 − 0.866i)5-s + (2 + 1.73i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s − 3·13-s + 1.99·15-s + (1 + 1.73i)17-s + (2.5 − 4.33i)19-s + (−0.999 + 5.19i)21-s + (−3.5 + 6.06i)23-s + (−0.499 − 0.866i)25-s + 4.00·27-s − 6·29-s + (−2 − 3.46i)31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.999i)3-s + (0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s + (−0.166 + 0.288i)9-s + (0.150 + 0.261i)11-s − 0.832·13-s + 0.516·15-s + (0.242 + 0.420i)17-s + (0.573 − 0.993i)19-s + (−0.218 + 1.13i)21-s + (−0.729 + 1.26i)23-s + (−0.0999 − 0.173i)25-s + 0.769·27-s − 1.11·29-s + (−0.359 − 0.622i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47618 + 0.731782i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47618 + 0.731782i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−11.90916132068886076602263077196, −11.04170892854981123514812127054, −9.709956987895203544572226384763, −9.400965746565245285952226948105, −8.365446723728180019029697061970, −7.32639241457906119462065740673, −5.63592321864802548724710202507, −4.79983762453541628848999220631, −3.65215614499862706658150815344, −2.10751769670711864828358578601,
1.50823482844571295798169859551, 2.81263459403753428472359451102, 4.40508231038512668784518427049, 5.84577941115573122685194343304, 7.17312428206905748195736345130, 7.63342710152771462573271871682, 8.601295883012427719679500758271, 9.903969746284081099909220949965, 10.75844807541013609832525979028, 11.88863007306304188272456430224