| L(s) = 1 | + 0.752·2-s + 2.90·3-s − 1.43·4-s + 0.247·5-s + 2.18·6-s − 2.58·8-s + 5.43·9-s + 0.186·10-s − 11-s − 4.16·12-s − 0.470·13-s + 0.717·15-s + 0.921·16-s + 17-s + 4.09·18-s − 6.84·19-s − 0.354·20-s − 0.752·22-s − 2.40·23-s − 7.50·24-s − 4.93·25-s − 0.354·26-s + 7.06·27-s − 0.186·29-s + 0.540·30-s + 7.93·31-s + 5.86·32-s + ⋯ |
| L(s) = 1 | + 0.532·2-s + 1.67·3-s − 0.716·4-s + 0.110·5-s + 0.892·6-s − 0.913·8-s + 1.81·9-s + 0.0588·10-s − 0.301·11-s − 1.20·12-s − 0.130·13-s + 0.185·15-s + 0.230·16-s + 0.242·17-s + 0.964·18-s − 1.56·19-s − 0.0792·20-s − 0.160·22-s − 0.502·23-s − 1.53·24-s − 0.987·25-s − 0.0694·26-s + 1.35·27-s − 0.0345·29-s + 0.0986·30-s + 1.42·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.992910333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992910333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.752T + 2T^{2} \) |
| 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 0.247T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 0.470T + 13T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 + 0.186T + 29T^{2} \) |
| 31 | \( 1 - 7.93T + 31T^{2} \) |
| 37 | \( 1 - 8.40T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 0.941T + 43T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 53 | \( 1 - 4.84T + 53T^{2} \) |
| 59 | \( 1 - 2.13T + 59T^{2} \) |
| 61 | \( 1 - 5.01T + 61T^{2} \) |
| 67 | \( 1 + 8.10T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 1.36T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−13.13735877909310916138042203210, −11.98597290042215667352570760002, −10.23352815266643296911422337386, −9.470930196742936925369865213035, −8.487008508931206141106382540497, −7.87938400078670010312002550095, −6.24861883288368621114843345168, −4.62361601214992219101191820679, −3.65916055117847043612387173182, −2.38975038235373592751925613524,
2.38975038235373592751925613524, 3.65916055117847043612387173182, 4.62361601214992219101191820679, 6.24861883288368621114843345168, 7.87938400078670010312002550095, 8.487008508931206141106382540497, 9.470930196742936925369865213035, 10.23352815266643296911422337386, 11.98597290042215667352570760002, 13.13735877909310916138042203210
