| L(s) = 1 | − 28.9·3-s − 146.·7-s + 594.·9-s + 191.·11-s − 83.9·13-s + 2.00e3·17-s + 677.·19-s + 4.24e3·21-s − 1.29e3·23-s − 1.01e4·27-s − 3.26e3·29-s + 6.15e3·31-s − 5.53e3·33-s + 1.13e4·37-s + 2.43e3·39-s − 1.05e4·41-s − 1.29e4·43-s − 9.52e3·47-s + 4.75e3·49-s − 5.78e4·51-s − 1.47e4·53-s − 1.96e4·57-s − 3.82e4·59-s − 3.58e3·61-s − 8.72e4·63-s − 2.17e4·67-s + 3.75e4·69-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 1.13·7-s + 2.44·9-s + 0.476·11-s − 0.137·13-s + 1.67·17-s + 0.430·19-s + 2.10·21-s − 0.510·23-s − 2.68·27-s − 0.721·29-s + 1.15·31-s − 0.883·33-s + 1.36·37-s + 0.255·39-s − 0.984·41-s − 1.06·43-s − 0.628·47-s + 0.282·49-s − 3.11·51-s − 0.723·53-s − 0.799·57-s − 1.42·59-s − 0.123·61-s − 2.76·63-s − 0.592·67-s + 0.948·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 28.9T + 243T^{2} \) |
| 7 | \( 1 + 146.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 191.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 83.9T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.00e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 677.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.52e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.08e4T + 8.58e9T^{2} \) |
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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
−11.25266686738373277126933776644, −10.04087707936115651467164633257, −9.686303972755360941995781004758, −7.71228941890068635770066459695, −6.56315340439722353109631382447, −5.95113505955021008928599689935, −4.88271155643483363714935411401, −3.48761282402639971760457609845, −1.19678378750398756566297678686, 0,
1.19678378750398756566297678686, 3.48761282402639971760457609845, 4.88271155643483363714935411401, 5.95113505955021008928599689935, 6.56315340439722353109631382447, 7.71228941890068635770066459695, 9.686303972755360941995781004758, 10.04087707936115651467164633257, 11.25266686738373277126933776644
