| L(s) = 1 | − 2.82·3-s + 17.4·5-s + 4.62·7-s − 19.0·9-s − 11·11-s + 85.1·13-s − 49.3·15-s − 6.08·17-s − 52.0·19-s − 13.0·21-s + 183.·23-s + 179.·25-s + 130.·27-s + 140.·29-s + 250.·31-s + 31.0·33-s + 80.7·35-s − 203.·37-s − 240.·39-s − 22.3·41-s + 117.·43-s − 331.·45-s + 275.·47-s − 321.·49-s + 17.2·51-s + 26.6·53-s − 191.·55-s + ⋯ |
| L(s) = 1 | − 0.543·3-s + 1.56·5-s + 0.249·7-s − 0.704·9-s − 0.301·11-s + 1.81·13-s − 0.849·15-s − 0.0868·17-s − 0.628·19-s − 0.135·21-s + 1.66·23-s + 1.43·25-s + 0.926·27-s + 0.899·29-s + 1.45·31-s + 0.164·33-s + 0.389·35-s − 0.904·37-s − 0.987·39-s − 0.0850·41-s + 0.416·43-s − 1.09·45-s + 0.855·47-s − 0.937·49-s + 0.0472·51-s + 0.0690·53-s − 0.470·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.909293197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909293197\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 2.82T + 27T^{2} \) |
| 5 | \( 1 - 17.4T + 125T^{2} \) |
| 7 | \( 1 - 4.62T + 343T^{2} \) |
| 13 | \( 1 - 85.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.08T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 203.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 22.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 117.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 26.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 515.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 693.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 341.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 831.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 917.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 456.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 91.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 146.T + 9.12e5T^{2} \) |
| 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
−12.26493374502580259856449175759, −10.95115408366757555441879919781, −10.55042054952550014121150604979, −9.159389746564876472758176330667, −8.414344364195272490030343555466, −6.53411566435038972780098326843, −5.94072332781738399418292534733, −4.88746715399639061302852648004, −2.87260602329607312531973249726, −1.25556648852584203839773216698,
1.25556648852584203839773216698, 2.87260602329607312531973249726, 4.88746715399639061302852648004, 5.94072332781738399418292534733, 6.53411566435038972780098326843, 8.414344364195272490030343555466, 9.159389746564876472758176330667, 10.55042054952550014121150604979, 10.95115408366757555441879919781, 12.26493374502580259856449175759
