L(s) = 1 | + 1.85·3-s + (−1.27 + 3.92i)5-s + (−0.809 − 0.587i)7-s + 0.451·9-s + (1.01 + 3.12i)11-s + (1.36 − 0.993i)13-s + (−2.37 + 7.29i)15-s + (1.46 + 4.51i)17-s + (−4.77 − 3.46i)19-s + (−1.50 − 1.09i)21-s + (3.94 − 2.86i)23-s + (−9.75 − 7.09i)25-s − 4.73·27-s + (−1.01 + 3.13i)29-s + (1.76 + 5.42i)31-s + ⋯ |
L(s) = 1 | + 1.07·3-s + (−0.570 + 1.75i)5-s + (−0.305 − 0.222i)7-s + 0.150·9-s + (0.305 + 0.941i)11-s + (0.379 − 0.275i)13-s + (−0.612 + 1.88i)15-s + (0.356 + 1.09i)17-s + (−1.09 − 0.795i)19-s + (−0.327 − 0.238i)21-s + (0.822 − 0.597i)23-s + (−1.95 − 1.41i)25-s − 0.911·27-s + (−0.189 + 0.581i)29-s + (0.316 + 0.975i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.638918065\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.638918065\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 41 | \( 1 + (3.90 - 5.07i)T \) |
good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 + (1.27 - 3.92i)T + (-4.04 - 2.93i)T^{2} \) |
| 11 | \( 1 + (-1.01 - 3.12i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 0.993i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.46 - 4.51i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.77 + 3.46i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.94 + 2.86i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.01 - 3.13i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.76 - 5.42i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.62 - 4.99i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (4.26 - 3.10i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-0.898 + 0.652i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.77 - 8.54i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.58 + 4.78i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.303 - 0.220i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.46 + 7.59i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.70 + 11.3i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + 2.91T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 - 7.52T + 83T^{2} \) |
| 89 | \( 1 + (-12.9 - 9.40i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.870 + 2.67i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.27181120986159670937606025264, −9.192772993711842058832799275581, −8.342959029894061222865647269312, −7.65762234992007579522153955888, −6.74641691445723415465331818416, −6.36905753723703373105868224413, −4.62905729876522000888764173811, −3.52005017440716018650072860226, −3.07929654636586668879756409870, −2.02666145205623098696942116729,
0.60631309318528062174177671192, 2.02836315135452487844322136308, 3.43257192209735042092211491526, 4.04032191404813906688904487210, 5.19762028383661550002429388462, 5.97409388482042298506263016138, 7.36984695869329107861570890099, 8.207014704507430227289954103052, 8.752660654290143493138726793320, 9.114867420801706743338192297512