| L(s) = 1 | + 4.65·3-s − 5·5-s + 7·7-s − 5.31·9-s + 52.2·11-s + 30.6·13-s − 23.2·15-s + 37.2·17-s − 80.2·19-s + 32.5·21-s − 25.8·23-s + 25·25-s − 150.·27-s + 20.9·29-s + 314.·31-s + 243.·33-s − 35·35-s + 197.·37-s + 142.·39-s + 11.3·41-s + 33.8·43-s + 26.5·45-s + 361.·47-s + 49·49-s + 173.·51-s + 153.·53-s − 261.·55-s + ⋯ |
| L(s) = 1 | + 0.896·3-s − 0.447·5-s + 0.377·7-s − 0.196·9-s + 1.43·11-s + 0.654·13-s − 0.400·15-s + 0.531·17-s − 0.968·19-s + 0.338·21-s − 0.234·23-s + 0.200·25-s − 1.07·27-s + 0.134·29-s + 1.82·31-s + 1.28·33-s − 0.169·35-s + 0.875·37-s + 0.586·39-s + 0.0432·41-s + 0.119·43-s + 0.0880·45-s + 1.12·47-s + 0.142·49-s + 0.475·51-s + 0.396·53-s − 0.640·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.834597996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.834597996\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 4.65T + 27T^{2} \) |
| 11 | \( 1 - 52.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 11.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 33.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 616T + 2.05e5T^{2} \) |
| 61 | \( 1 - 15.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 166.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 952T + 3.57e5T^{2} \) |
| 73 | \( 1 + 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 45.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
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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
−10.28106354695442536119364281249, −9.249509787807976425309992784933, −8.524353351140309678324763188754, −7.984567827427055002540320018741, −6.79548820583060812781754778118, −5.87775508944929476062047480217, −4.35857900253479350063769821294, −3.65145473247428242929094068901, −2.43721924732272623395284712248, −1.04039831827976666961139235274,
1.04039831827976666961139235274, 2.43721924732272623395284712248, 3.65145473247428242929094068901, 4.35857900253479350063769821294, 5.87775508944929476062047480217, 6.79548820583060812781754778118, 7.984567827427055002540320018741, 8.524353351140309678324763188754, 9.249509787807976425309992784933, 10.28106354695442536119364281249
