L(s) = 1 | − 6.98·2-s + 9·3-s + 16.7·4-s − 62.8·6-s − 180.·7-s + 106.·8-s + 81·9-s − 121·11-s + 151.·12-s + 723.·13-s + 1.25e3·14-s − 1.27e3·16-s + 1.49e3·17-s − 565.·18-s + 2.59e3·19-s − 1.62e3·21-s + 845.·22-s + 4.06e3·23-s + 956.·24-s − 5.05e3·26-s + 729·27-s − 3.02e3·28-s − 7.80e3·29-s − 49.3·31-s + 5.53e3·32-s − 1.08e3·33-s − 1.04e4·34-s + ⋯ |
L(s) = 1 | − 1.23·2-s + 0.577·3-s + 0.524·4-s − 0.712·6-s − 1.39·7-s + 0.587·8-s + 0.333·9-s − 0.301·11-s + 0.302·12-s + 1.18·13-s + 1.71·14-s − 1.24·16-s + 1.25·17-s − 0.411·18-s + 1.65·19-s − 0.803·21-s + 0.372·22-s + 1.60·23-s + 0.338·24-s − 1.46·26-s + 0.192·27-s − 0.729·28-s − 1.72·29-s − 0.00921·31-s + 0.955·32-s − 0.174·33-s − 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.252380915\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.252380915\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 + 6.98T + 32T^{2} \) |
| 7 | \( 1 + 180.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 723.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.59e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 49.3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.78e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.47e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.70e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.77e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 743.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.68e5T + 8.58e9T^{2} \) |
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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
−9.399715661323209553501497604722, −8.896651449927714972987093444741, −7.80441703353788334075448552135, −7.31740367865106202538553620559, −6.26203795693121679595360618480, −5.15109388200906050237050121547, −3.58708925607332497438686201906, −3.05197030105533413510084669213, −1.48762352357359313260213413837, −0.64616204134501543845890674432,
0.64616204134501543845890674432, 1.48762352357359313260213413837, 3.05197030105533413510084669213, 3.58708925607332497438686201906, 5.15109388200906050237050121547, 6.26203795693121679595360618480, 7.31740367865106202538553620559, 7.80441703353788334075448552135, 8.896651449927714972987093444741, 9.399715661323209553501497604722