| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s − 13-s + 14-s + 15-s + 16-s + 4·17-s + 18-s − 8·19-s + 20-s + 21-s − 2·22-s + 3·23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.16559121812652, −14.91839307838982, −14.40339396656486, −14.11360156375573, −13.15268701397524, −12.94808649141712, −12.67987938004368, −11.75699166460635, −11.28161145894927, −10.51676280933675, −10.32306238783061, −9.510512232470790, −8.949311741987934, −8.242231391161480, −7.883953489710594, −6.977490803303436, −6.785099073190458, −5.732995225599786, −5.359595189057849, −4.791105718692056, −3.933502017950845, −3.484662923471325, −2.625340496764553, −2.040825247149687, −1.444846517229007, 0,
1.444846517229007, 2.040825247149687, 2.625340496764553, 3.484662923471325, 3.933502017950845, 4.791105718692056, 5.359595189057849, 5.732995225599786, 6.785099073190458, 6.977490803303436, 7.883953489710594, 8.242231391161480, 8.949311741987934, 9.510512232470790, 10.32306238783061, 10.51676280933675, 11.28161145894927, 11.75699166460635, 12.67987938004368, 12.94808649141712, 13.15268701397524, 14.11360156375573, 14.40339396656486, 14.91839307838982, 15.16559121812652
