L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 + 1.73i)5-s + (2 + 3.46i)7-s + 0.999·8-s + 1.99·10-s + (0.5 + 0.866i)11-s + (3 − 5.19i)13-s + (1.99 − 3.46i)14-s + (−0.5 − 0.866i)16-s + 2·17-s + 4·19-s + (−0.999 − 1.73i)20-s + (0.499 − 0.866i)22-s + (−2 + 3.46i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.755 + 1.30i)7-s + 0.353·8-s + 0.632·10-s + (0.150 + 0.261i)11-s + (0.832 − 1.44i)13-s + (0.534 − 0.925i)14-s + (−0.125 − 0.216i)16-s + 0.485·17-s + 0.917·19-s + (−0.223 − 0.387i)20-s + (0.106 − 0.184i)22-s + (−0.417 + 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.469590040\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.469590040\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.343205206338520900407985765824, −8.662491175949813007331539266276, −7.75383918100423541242416759766, −7.45892180871969647440146292828, −5.87026927332905148122852790369, −5.52683411646643825956499354129, −4.17729392851415610741626357656, −3.19444863872429356283236453158, −2.51016141691154590015405160160, −1.21137917161860257271172838558,
0.76445687512027243534363527773, 1.61974026695972400219071448857, 3.61561688605682570263444616910, 4.37223675632134724812614038870, 4.98084963502344651655389422168, 6.14359628000179431426173367395, 6.96137039359423436323101648569, 7.70165859237757295608375076862, 8.325251848935204919982984448834, 9.016091503447965786795765476800