| L(s) = 1 | − 5.01·3-s − 15.3·7-s − 217.·9-s + 576.·11-s + 607.·13-s + 2.01e3·17-s + 2.42e3·19-s + 77.2·21-s − 4.35e3·23-s + 2.31e3·27-s + 1.22e3·29-s − 7.81e3·31-s − 2.89e3·33-s − 3.80e3·37-s − 3.04e3·39-s − 4.47e3·41-s − 1.47e4·43-s + 2.04e4·47-s − 1.65e4·49-s − 1.01e4·51-s + 1.09e4·53-s − 1.21e4·57-s + 3.11e4·59-s − 4.48e3·61-s + 3.35e3·63-s + 4.54e4·67-s + 2.18e4·69-s + ⋯ |
| L(s) = 1 | − 0.321·3-s − 0.118·7-s − 0.896·9-s + 1.43·11-s + 0.996·13-s + 1.68·17-s + 1.53·19-s + 0.0382·21-s − 1.71·23-s + 0.610·27-s + 0.270·29-s − 1.46·31-s − 0.462·33-s − 0.457·37-s − 0.320·39-s − 0.415·41-s − 1.21·43-s + 1.35·47-s − 0.985·49-s − 0.543·51-s + 0.533·53-s − 0.495·57-s + 1.16·59-s − 0.154·61-s + 0.106·63-s + 1.23·67-s + 0.552·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.175783577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.175783577\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.01T + 243T^{2} \) |
| 7 | \( 1 + 15.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 576.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 607.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.42e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.47e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.48e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.54e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.89e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−9.543868277327180397515652135750, −8.680495913044203319534912014737, −7.86723118186159339159251277285, −6.81368791599597877318830334561, −5.88011792481343880904018968614, −5.37264601200789563469899384578, −3.81729855836879721576481551728, −3.30076372927221460831916108489, −1.66400447897510956979902835740, −0.72365262250957828911683125123,
0.72365262250957828911683125123, 1.66400447897510956979902835740, 3.30076372927221460831916108489, 3.81729855836879721576481551728, 5.37264601200789563469899384578, 5.88011792481343880904018968614, 6.81368791599597877318830334561, 7.86723118186159339159251277285, 8.680495913044203319534912014737, 9.543868277327180397515652135750
