| L(s) = 1 | + 4·2-s + 8.19·3-s + 16·4-s − 66.8·5-s + 32.7·6-s − 215.·7-s + 64·8-s − 175.·9-s − 267.·10-s + 509.·11-s + 131.·12-s + 868.·13-s − 862.·14-s − 547.·15-s + 256·16-s + 2.23e3·17-s − 703.·18-s + 1.26e3·19-s − 1.06e3·20-s − 1.76e3·21-s + 2.03e3·22-s − 2.23e3·23-s + 524.·24-s + 1.33e3·25-s + 3.47e3·26-s − 3.43e3·27-s − 3.45e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.525·3-s + 0.5·4-s − 1.19·5-s + 0.371·6-s − 1.66·7-s + 0.353·8-s − 0.723·9-s − 0.845·10-s + 1.26·11-s + 0.262·12-s + 1.42·13-s − 1.17·14-s − 0.628·15-s + 0.250·16-s + 1.87·17-s − 0.511·18-s + 0.801·19-s − 0.597·20-s − 0.874·21-s + 0.897·22-s − 0.880·23-s + 0.185·24-s + 0.428·25-s + 1.00·26-s − 0.905·27-s − 0.831·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 431 | \( 1 - 1.85e5T \) |
| good | 3 | \( 1 - 8.19T + 243T^{2} \) |
| 5 | \( 1 + 66.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 215.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 509.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 868.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.23e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.33e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.38e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.23e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.64e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.63e3T + 8.58e9T^{2} \) |
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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
−9.100274576969199469010398777541, −7.962218303105610524103644459709, −7.36020780641800976926715055080, −6.14622435984285386700638473916, −5.77914460798531190987128142467, −3.93080827593253631941426681961, −3.59135011754795182825528955734, −3.05188573207374799444967371207, −1.29621288745759408052932127115, 0,
1.29621288745759408052932127115, 3.05188573207374799444967371207, 3.59135011754795182825528955734, 3.93080827593253631941426681961, 5.77914460798531190987128142467, 6.14622435984285386700638473916, 7.36020780641800976926715055080, 7.962218303105610524103644459709, 9.100274576969199469010398777541
