L(s) = 1 | − 1.50·3-s − 0.787·5-s − 27.3·7-s − 24.7·9-s − 62.2·11-s + 27.7·13-s + 1.18·15-s + 102.·17-s − 55.9·19-s + 41.0·21-s − 175.·23-s − 124.·25-s + 77.6·27-s + 29·29-s − 199.·31-s + 93.3·33-s + 21.5·35-s − 403.·37-s − 41.7·39-s − 118.·41-s − 354.·43-s + 19.4·45-s + 148.·47-s + 405.·49-s − 153.·51-s − 376.·53-s + 49.0·55-s + ⋯ |
L(s) = 1 | − 0.288·3-s − 0.0704·5-s − 1.47·7-s − 0.916·9-s − 1.70·11-s + 0.592·13-s + 0.0203·15-s + 1.46·17-s − 0.675·19-s + 0.426·21-s − 1.59·23-s − 0.995·25-s + 0.553·27-s + 0.185·29-s − 1.15·31-s + 0.492·33-s + 0.103·35-s − 1.79·37-s − 0.171·39-s − 0.450·41-s − 1.25·43-s + 0.0645·45-s + 0.461·47-s + 1.18·49-s − 0.422·51-s − 0.974·53-s + 0.120·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.03935080898\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03935080898\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
good | 3 | \( 1 + 1.50T + 27T^{2} \) |
| 5 | \( 1 + 0.787T + 125T^{2} \) |
| 7 | \( 1 + 27.3T + 343T^{2} \) |
| 11 | \( 1 + 62.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 27.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 403.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 148.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 376.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 329.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 789.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 147.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 254.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 402.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 429.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 78.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3T + 9.12e5T^{2} \) |
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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
−8.829206865687851535890025499421, −8.070700646408093526393605718381, −7.39961850838928857164867664040, −6.16874989587360189189735780085, −5.87972617036150276090504177091, −5.02101368721346735216572094752, −3.56475722374994564352265553138, −3.14500180698528377728118099750, −1.95740785947200604681606788675, −0.087061731208728989335388871783,
0.087061731208728989335388871783, 1.95740785947200604681606788675, 3.14500180698528377728118099750, 3.56475722374994564352265553138, 5.02101368721346735216572094752, 5.87972617036150276090504177091, 6.16874989587360189189735780085, 7.39961850838928857164867664040, 8.070700646408093526393605718381, 8.829206865687851535890025499421