| L(s) = 1 | + (5.26 + 9.11i)5-s + (0.177 − 0.307i)7-s + (−4.30 + 7.46i)11-s + (17.3 + 30.1i)13-s + 72.5·17-s − 78.3·19-s + (24.6 + 42.6i)23-s + (7.15 − 12.3i)25-s + (−69.4 + 120. i)29-s + (31.2 + 54.0i)31-s + 3.73·35-s + 36.0·37-s + (−95.4 − 165. i)41-s + (−266. + 461. i)43-s + (35.3 − 61.2i)47-s + ⋯ |
| L(s) = 1 | + (0.470 + 0.814i)5-s + (0.00958 − 0.0165i)7-s + (−0.118 + 0.204i)11-s + (0.371 + 0.642i)13-s + 1.03·17-s − 0.946·19-s + (0.223 + 0.386i)23-s + (0.0572 − 0.0991i)25-s + (−0.444 + 0.769i)29-s + (0.180 + 0.313i)31-s + 0.0180·35-s + 0.160·37-s + (−0.363 − 0.630i)41-s + (−0.944 + 1.63i)43-s + (0.109 − 0.189i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.759185479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759185479\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-5.26 - 9.11i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-0.177 + 0.307i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (4.30 - 7.46i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-17.3 - 30.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 72.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 78.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-24.6 - 42.6i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (69.4 - 120. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-31.2 - 54.0i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 36.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + (95.4 + 165. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (266. - 461. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-35.3 + 61.2i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 245.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (417. + 723. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (240. - 416. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (55.2 + 95.7i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 58.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 313.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (497. - 861. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (366. - 635. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 994.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (529. - 916. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−10.44740037535785299253597384819, −9.661806887619352271071836250990, −8.747057426922645863130695335835, −7.71358838495399061590545952701, −6.77745814163484711934449353541, −6.08263370107097271915357690278, −4.96825354856421125050361905389, −3.73531476088355184780261517786, −2.66551327601651172190024736689, −1.44993464511246985129412115651,
0.50530567120273383673385495752, 1.74444643082687577671229240010, 3.13767047597355984370778350442, 4.36342223987168559892731444388, 5.42324381373964660602005079190, 6.06009399961022764517004888693, 7.31877713873825340841662433827, 8.325736558582798178174308741548, 8.908419365896068376720045666473, 9.953379852122065175509919178522
