| L(s) = 1 | + (−3.23 + 2.34i)3-s + (−2.54 − 7.82i)5-s + (1.55 − 2.14i)7-s + (2.15 − 6.63i)9-s + (3.85 + 10.3i)11-s + (4.32 + 1.40i)13-s + (26.6 + 19.3i)15-s + (−23.7 + 7.71i)17-s + (6.86 + 9.44i)19-s + 10.5i·21-s + 9.61·23-s + (−34.5 + 25.0i)25-s + (−2.49 − 7.68i)27-s + (−22.3 + 30.7i)29-s + (−7.53 + 23.1i)31-s + ⋯ |
| L(s) = 1 | + (−1.07 + 0.783i)3-s + (−0.508 − 1.56i)5-s + (0.222 − 0.305i)7-s + (0.239 − 0.737i)9-s + (0.350 + 0.936i)11-s + (0.332 + 0.107i)13-s + (1.77 + 1.28i)15-s + (−1.39 + 0.453i)17-s + (0.361 + 0.497i)19-s + 0.503i·21-s + 0.418·23-s + (−1.38 + 1.00i)25-s + (−0.0924 − 0.284i)27-s + (−0.769 + 1.05i)29-s + (−0.243 + 0.748i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.337206 + 0.474757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.337206 + 0.474757i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.55 + 2.14i)T \) |
| 11 | \( 1 + (-3.85 - 10.3i)T \) |
| good | 3 | \( 1 + (3.23 - 2.34i)T + (2.78 - 8.55i)T^{2} \) |
| 5 | \( 1 + (2.54 + 7.82i)T + (-20.2 + 14.6i)T^{2} \) |
| 13 | \( 1 + (-4.32 - 1.40i)T + (136. + 99.3i)T^{2} \) |
| 17 | \( 1 + (23.7 - 7.71i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-6.86 - 9.44i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 - 9.61T + 529T^{2} \) |
| 29 | \( 1 + (22.3 - 30.7i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (7.53 - 23.1i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-36.8 - 26.7i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-5.08 - 6.99i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 23.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (45.5 - 33.1i)T + (682. - 2.10e3i)T^{2} \) |
| 53 | \( 1 + (6.50 - 20.0i)T + (-2.27e3 - 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-46.1 - 33.5i)T + (1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-44.7 + 14.5i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 - 70.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-35.2 - 108. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-17.7 + 24.3i)T + (-1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (107. + 34.8i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-70.4 + 22.8i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + 140.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-39.8 + 122. i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.59632205046330477328855139690, −11.04438050448714439930947713403, −9.897211639656402367200764635626, −9.050092120499190958398785147531, −8.108119267387091251975102503643, −6.77253272331720061148425502875, −5.45630342811819523727151233338, −4.64716933711665638336727057331, −4.06622744939206235539226049754, −1.35043555361785785574820177540,
0.34792286927126442746475702961, 2.42071828489384388810238238967, 3.80952501815611706930504116422, 5.51030714736311773314371007074, 6.46520434635826794735663908325, 6.98617840948018637283786572613, 8.039941181962952586640795476184, 9.369263825525593271556014415152, 10.84046088020109546764577061481, 11.36061166967980027417886908978
