| L(s) = 1 | + 2.92·2-s − 23.4·4-s − 25·5-s − 85.0·7-s − 162.·8-s − 73.0·10-s + 121·11-s + 724.·13-s − 248.·14-s + 277.·16-s + 2.09e3·17-s − 6.40·19-s + 586.·20-s + 353.·22-s − 1.56e3·23-s + 625·25-s + 2.11e3·26-s + 1.99e3·28-s + 5.14e3·29-s − 1.03e3·31-s + 5.99e3·32-s + 6.13e3·34-s + 2.12e3·35-s − 1.26e4·37-s − 18.7·38-s + 4.05e3·40-s − 1.38e4·41-s + ⋯ |
| L(s) = 1 | + 0.516·2-s − 0.733·4-s − 0.447·5-s − 0.655·7-s − 0.895·8-s − 0.230·10-s + 0.301·11-s + 1.18·13-s − 0.338·14-s + 0.270·16-s + 1.76·17-s − 0.00406·19-s + 0.327·20-s + 0.155·22-s − 0.618·23-s + 0.200·25-s + 0.613·26-s + 0.480·28-s + 1.13·29-s − 0.192·31-s + 1.03·32-s + 0.909·34-s + 0.293·35-s − 1.51·37-s − 0.00210·38-s + 0.400·40-s − 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 2.92T + 32T^{2} \) |
| 7 | \( 1 + 85.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 724.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 6.40T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.07e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5T + 8.58e9T^{2} \) |
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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
−9.777563301270234482142518988416, −8.702319673372563060516155202671, −8.100518724128599711542208575022, −6.74925773487906141427605329284, −5.86098245881673664806125340438, −4.91759293040884075549980072283, −3.65324627213714204116445865276, −3.29710285536882667458004093649, −1.24865251427612106427304733822, 0,
1.24865251427612106427304733822, 3.29710285536882667458004093649, 3.65324627213714204116445865276, 4.91759293040884075549980072283, 5.86098245881673664806125340438, 6.74925773487906141427605329284, 8.100518724128599711542208575022, 8.702319673372563060516155202671, 9.777563301270234482142518988416
