| L(s) = 1 | − 5.29·3-s − 10.5·5-s + 8·7-s + 1.00·9-s − 15.8·11-s − 52.9·13-s + 56.0·15-s − 14·17-s + 37.0·19-s − 42.3·21-s + 152·23-s − 12.9·25-s + 137.·27-s + 158.·29-s + 224·31-s + 84.0·33-s − 84.6·35-s + 243.·37-s + 280·39-s + 70·41-s − 439.·43-s − 10.5·45-s + 336·47-s − 279·49-s + 74.0·51-s + 31.7·53-s + 168.·55-s + ⋯ |
| L(s) = 1 | − 1.01·3-s − 0.946·5-s + 0.431·7-s + 0.0370·9-s − 0.435·11-s − 1.12·13-s + 0.963·15-s − 0.199·17-s + 0.447·19-s − 0.439·21-s + 1.37·23-s − 0.103·25-s + 0.980·27-s + 1.01·29-s + 1.29·31-s + 0.443·33-s − 0.408·35-s + 1.08·37-s + 1.14·39-s + 0.266·41-s − 1.55·43-s − 0.0350·45-s + 1.04·47-s − 0.813·49-s + 0.203·51-s + 0.0822·53-s + 0.411·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7899003358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7899003358\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 5.29T + 27T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 - 8T + 343T^{2} \) |
| 11 | \( 1 + 15.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 70T + 6.89e4T^{2} \) |
| 43 | \( 1 + 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 336T + 1.03e5T^{2} \) |
| 53 | \( 1 - 31.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 534.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 95.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 174.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 72T + 3.57e5T^{2} \) |
| 73 | \( 1 - 294T + 3.89e5T^{2} \) |
| 79 | \( 1 + 464T + 4.93e5T^{2} \) |
| 83 | \( 1 - 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994T + 9.12e5T^{2} \) |
| 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
−11.66933170036289674095422955238, −10.87046686985733600985108456487, −9.879830724625259311823062510702, −8.488634355589628470945753081801, −7.58178639920647970867712128052, −6.57749887798527065367562507660, −5.23290807731858407769584417162, −4.54468078533690786678526468450, −2.84465829878519984602927899742, −0.66043675447049262693983069160,
0.66043675447049262693983069160, 2.84465829878519984602927899742, 4.54468078533690786678526468450, 5.23290807731858407769584417162, 6.57749887798527065367562507660, 7.58178639920647970867712128052, 8.488634355589628470945753081801, 9.879830724625259311823062510702, 10.87046686985733600985108456487, 11.66933170036289674095422955238
