L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (2.5 + 4.33i)5-s + (−7 + 12.1i)7-s − 7.99·8-s + 10·10-s + (−3 + 5.19i)11-s + (−34 − 58.8i)13-s + (14 + 24.2i)14-s + (−8 + 13.8i)16-s + 78·17-s + 44·19-s + (10 − 17.3i)20-s + (6 + 10.3i)22-s + (−60 − 103. i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.377 + 0.654i)7-s − 0.353·8-s + 0.316·10-s + (−0.0822 + 0.142i)11-s + (−0.725 − 1.25i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s + 1.11·17-s + 0.531·19-s + (0.111 − 0.193i)20-s + (0.0581 + 0.100i)22-s + (−0.543 − 0.942i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.189582651\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.189582651\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
good | 7 | \( 1 + (7 - 12.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (34 + 58.8i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 78T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44T + 6.85e3T^{2} \) |
| 23 | \( 1 + (60 + 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (63 - 109. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-122 - 211. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 304T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-240 - 415. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (52 - 90.0i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (300 - 519. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 258T + 1.48e5T^{2} \) |
| 59 | \( 1 + (267 + 462. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (181 - 313. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-134 - 232. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 972T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470T + 3.89e5T^{2} \) |
| 79 | \( 1 + (622 - 1.07e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (198 - 342. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 972T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-23 + 39.8i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.07515962067977572533162886955, −9.545811076780990486209945250317, −8.401728481077574997744826076223, −7.51340396021370086516600660038, −6.34557156976400250152855756372, −5.53635420023510681536322881769, −4.72993058818170728910832964292, −3.17968935316439485258884916323, −2.79179759940288663010719156394, −1.30458702128427782373890446337,
0.28124779898681569155790539712, 1.88304406892896568762972992588, 3.42572469535070706395919132657, 4.26999220691637587166254516810, 5.30195790978287636700409965502, 6.10101231172062599908926530671, 7.18414038448278547527218492124, 7.66864431526888342118215882008, 8.821109965430760364129334379176, 9.654815656473822269402845699659