| L(s) = 1 | − 1.17·3-s − 5-s + 7-s − 1.63·9-s + 11-s − 0.0917·13-s + 1.17·15-s − 5.51·17-s − 0.921·19-s − 1.17·21-s + 5.70·23-s + 25-s + 5.41·27-s + 1.41·29-s − 0.879·31-s − 1.17·33-s − 35-s − 8.78·37-s + 0.107·39-s − 1.61·41-s − 3.86·43-s + 1.63·45-s + 5.90·47-s + 49-s + 6.44·51-s − 10.0·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.675·3-s − 0.447·5-s + 0.377·7-s − 0.543·9-s + 0.301·11-s − 0.0254·13-s + 0.302·15-s − 1.33·17-s − 0.211·19-s − 0.255·21-s + 1.19·23-s + 0.200·25-s + 1.04·27-s + 0.263·29-s − 0.157·31-s − 0.203·33-s − 0.169·35-s − 1.44·37-s + 0.0171·39-s − 0.252·41-s − 0.589·43-s + 0.243·45-s + 0.861·47-s + 0.142·49-s + 0.902·51-s − 1.38·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9665450399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9665450399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.17T + 3T^{2} \) |
| 13 | \( 1 + 0.0917T + 13T^{2} \) |
| 17 | \( 1 + 5.51T + 17T^{2} \) |
| 19 | \( 1 + 0.921T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 0.879T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 + 3.86T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 - 2.14T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.52T + 83T^{2} \) |
| 89 | \( 1 + 2.83T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.145961689584727310029677439269, −7.16676730592835296698325442678, −6.68329336727894721517171745554, −5.96461999301644116978760923376, −5.04990805196482844759591341334, −4.66718377127495730440738357178, −3.67661049455491034417913213075, −2.81254219942022434827443307833, −1.75836927083405236663651483848, −0.52645517755485794879484103224,
0.52645517755485794879484103224, 1.75836927083405236663651483848, 2.81254219942022434827443307833, 3.67661049455491034417913213075, 4.66718377127495730440738357178, 5.04990805196482844759591341334, 5.96461999301644116978760923376, 6.68329336727894721517171745554, 7.16676730592835296698325442678, 8.145961689584727310029677439269
