| L(s) = 1 | + (0.913 + 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.122 + 1.16i)5-s + (−0.809 − 0.587i)6-s + (2.38 − 1.13i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.583 + 1.01i)10-s + (−0.0814 + 3.31i)11-s + (−0.499 − 0.866i)12-s + (0.534 − 0.388i)13-s + (2.64 − 0.0666i)14-s + (0.360 − 1.11i)15-s + (−0.104 + 0.994i)16-s + (−0.208 + 0.0927i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (−0.564 − 0.120i)3-s + (0.334 + 0.371i)4-s + (−0.0545 + 0.519i)5-s + (−0.330 − 0.239i)6-s + (0.903 − 0.429i)7-s + (0.109 + 0.336i)8-s + (0.304 + 0.135i)9-s + (−0.184 + 0.319i)10-s + (−0.0245 + 0.999i)11-s + (−0.144 − 0.249i)12-s + (0.148 − 0.107i)13-s + (0.706 − 0.0178i)14-s + (0.0931 − 0.286i)15-s + (−0.0261 + 0.248i)16-s + (−0.0505 + 0.0224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63768 + 0.853307i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63768 + 0.853307i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.913 - 0.406i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (-2.38 + 1.13i)T \) |
| 11 | \( 1 + (0.0814 - 3.31i)T \) |
| good | 5 | \( 1 + (0.122 - 1.16i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (-0.534 + 0.388i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.208 - 0.0927i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.261 + 0.290i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-4.34 - 7.53i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.834 + 2.56i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.0194 + 0.184i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (3.79 - 0.806i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (0.155 + 0.477i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4.81T + 43T^{2} \) |
| 47 | \( 1 + (0.773 - 0.858i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (0.977 + 9.30i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (7.01 + 7.79i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.994 + 9.46i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.15 + 3.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 3.89i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.80 + 10.8i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.103 - 0.0460i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (8.46 + 6.15i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-6.30 - 10.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.08 + 4.41i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.27262681848873328235379152322, −10.60880984994123943590570292906, −9.533020761094821019431771193861, −8.115813028614594836098298892911, −7.27543001200922032253114506570, −6.64545340465576959168475379561, −5.32874268622474867227619473976, −4.66615415774386678682763327597, −3.40203938006608013661913853846, −1.74491602648794809978860055311,
1.17014399180767496402746798176, 2.84202005723274065289430416704, 4.33122253092514978768055114961, 5.08976738996143766423878193525, 5.92581531329074529861239184570, 6.99589870180383512625038036935, 8.393674806450712659352805579495, 8.980361455584424397394144688369, 10.46981918128065341785337832257, 10.99000493631473827570841950001
