| L(s) = 1 | − 17.1·5-s − 23.1·7-s + 11·11-s − 59.6·13-s + 80.8·17-s − 11.7·19-s + 56.0·23-s + 168.·25-s + 85.4·29-s − 99.8·31-s + 396.·35-s + 402.·37-s − 27.0·41-s − 74·43-s + 408.·47-s + 192.·49-s − 463.·53-s − 188.·55-s + 498.·59-s − 635.·61-s + 1.02e3·65-s − 701.·67-s − 27.7·71-s + 619.·73-s − 254.·77-s − 208.·79-s + 1.30e3·83-s + ⋯ |
| L(s) = 1 | − 1.53·5-s − 1.24·7-s + 0.301·11-s − 1.27·13-s + 1.15·17-s − 0.141·19-s + 0.508·23-s + 1.34·25-s + 0.547·29-s − 0.578·31-s + 1.91·35-s + 1.79·37-s − 0.103·41-s − 0.262·43-s + 1.26·47-s + 0.560·49-s − 1.20·53-s − 0.462·55-s + 1.10·59-s − 1.33·61-s + 1.95·65-s − 1.27·67-s − 0.0463·71-s + 0.993·73-s − 0.376·77-s − 0.296·79-s + 1.72·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8419981721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8419981721\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 5 | \( 1 + 17.1T + 125T^{2} \) |
| 7 | \( 1 + 23.1T + 343T^{2} \) |
| 13 | \( 1 + 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 99.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 27.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 74T + 7.95e4T^{2} \) |
| 47 | \( 1 - 408.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 635.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 701.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 27.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 619.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 208.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91982989565892704612692699859, −9.899116144344225175505908477532, −9.121857091030088711357140205550, −7.85873772502926779614755362598, −7.29685703980196828688710092448, −6.23939274757616566235218479141, −4.82780199691401233358285965346, −3.73955158079746682598293345989, −2.86788086209098803791946835368, −0.57891263327620278241286908249,
0.57891263327620278241286908249, 2.86788086209098803791946835368, 3.73955158079746682598293345989, 4.82780199691401233358285965346, 6.23939274757616566235218479141, 7.29685703980196828688710092448, 7.85873772502926779614755362598, 9.121857091030088711357140205550, 9.899116144344225175505908477532, 10.91982989565892704612692699859
