L(s) = 1 | + (−0.895 + 1.55i)5-s + (2.08 − 1.20i)7-s + (−1.36 + 0.790i)11-s + (5.35 + 3.09i)13-s − 3.69i·17-s − 3.12·19-s + (−1.36 + 2.35i)23-s + (0.896 + 1.55i)25-s + (2.55 + 4.42i)29-s + (5.95 + 3.43i)31-s + 4.31i·35-s − 5.24i·37-s + (5.32 + 3.07i)41-s + (0.452 + 0.783i)43-s + (4.88 + 8.46i)47-s + ⋯ |
L(s) = 1 | + (−0.400 + 0.693i)5-s + (0.789 − 0.455i)7-s + (−0.412 + 0.238i)11-s + (1.48 + 0.857i)13-s − 0.897i·17-s − 0.717·19-s + (−0.283 + 0.491i)23-s + (0.179 + 0.310i)25-s + (0.474 + 0.821i)29-s + (1.06 + 0.617i)31-s + 0.729i·35-s − 0.861i·37-s + (0.831 + 0.479i)41-s + (0.0689 + 0.119i)43-s + (0.713 + 1.23i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46199 + 0.613590i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46199 + 0.613590i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.895 - 1.55i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.08 + 1.20i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.36 - 0.790i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.35 - 3.09i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.69iT - 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + (1.36 - 2.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.55 - 4.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.95 - 3.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.24iT - 37T^{2} \) |
| 41 | \( 1 + (-5.32 - 3.07i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.452 - 0.783i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.88 - 8.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 + (6.10 + 3.52i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.05 - 1.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.03 - 1.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 - 0.631T + 73T^{2} \) |
| 79 | \( 1 + (7.82 - 4.51i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-13.5 + 7.82i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.16iT - 89T^{2} \) |
| 97 | \( 1 + (6.72 + 11.6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.59468472415652896916303873882, −9.369785230842706522208800682747, −8.532816602502810369517542767334, −7.65609093104308507773358759466, −6.94501611589746615974609378192, −6.04020177694860182892032795946, −4.77791079368304432516264770692, −3.97199800654426282021842898348, −2.82536900019274867268142216316, −1.36804290151906425312233410114,
0.903400282097783585186406573534, 2.36808387569057283902430706220, 3.80698010878224349803744831040, 4.65149750603497527187710452812, 5.70256764683152006886588905435, 6.38950762331984734963074891103, 8.001139799866527783749097618482, 8.269155543679860225835650720053, 8.877747965419978682663210685971, 10.28939420724477650034140917365