| L(s) = 1 | − 3.66·2-s + 2.67·3-s + 5.41·4-s + 4.26·5-s − 9.78·6-s + 14.5·7-s + 9.48·8-s − 19.8·9-s − 15.6·10-s + 14.0·11-s + 14.4·12-s − 38.2·13-s − 53.4·14-s + 11.3·15-s − 78.0·16-s + 45.6·17-s + 72.7·18-s − 103.·19-s + 23.0·20-s + 38.9·21-s − 51.5·22-s + 25.3·24-s − 106.·25-s + 140.·26-s − 125.·27-s + 78.9·28-s − 149.·29-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.514·3-s + 0.676·4-s + 0.381·5-s − 0.665·6-s + 0.787·7-s + 0.419·8-s − 0.735·9-s − 0.493·10-s + 0.385·11-s + 0.347·12-s − 0.815·13-s − 1.01·14-s + 0.195·15-s − 1.21·16-s + 0.650·17-s + 0.952·18-s − 1.25·19-s + 0.257·20-s + 0.404·21-s − 0.499·22-s + 0.215·24-s − 0.854·25-s + 1.05·26-s − 0.892·27-s + 0.532·28-s − 0.955·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + 3.66T + 8T^{2} \) |
| 3 | \( 1 - 2.67T + 27T^{2} \) |
| 5 | \( 1 - 4.26T + 125T^{2} \) |
| 7 | \( 1 - 14.5T + 343T^{2} \) |
| 11 | \( 1 - 14.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 45.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 386.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 417.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 405.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 37.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 40.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 395.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 462.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 467.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 359.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 907.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 63.4T + 9.12e5T^{2} \) |
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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
−9.696368073890008487972836569988, −9.175283727945126199677104330298, −8.166692524643966020935818507618, −7.84660236276983967808059657022, −6.60336389231718816232557479288, −5.36673930143544557826218588306, −4.16223578400603672567340062942, −2.51871518683166273232826164581, −1.56578691179796660372121397522, 0,
1.56578691179796660372121397522, 2.51871518683166273232826164581, 4.16223578400603672567340062942, 5.36673930143544557826218588306, 6.60336389231718816232557479288, 7.84660236276983967808059657022, 8.166692524643966020935818507618, 9.175283727945126199677104330298, 9.696368073890008487972836569988
