L(s) = 1 | − 2-s + (−1.26 + 1.18i)3-s + 4-s + 3.37i·5-s + (1.26 − 1.18i)6-s + (2.52 − 0.792i)7-s − 8-s + (0.186 − 2.99i)9-s − 3.37i·10-s + 5.74·11-s + (−1.26 + 1.18i)12-s + (3.46 − i)13-s + (−2.52 + 0.792i)14-s + (−4 − 4.25i)15-s + 16-s + 0.792·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.728 + 0.684i)3-s + 0.5·4-s + 1.50i·5-s + (0.515 − 0.484i)6-s + (0.954 − 0.299i)7-s − 0.353·8-s + (0.0620 − 0.998i)9-s − 1.06i·10-s + 1.73·11-s + (−0.364 + 0.342i)12-s + (0.960 − 0.277i)13-s + (−0.674 + 0.211i)14-s + (−1.03 − 1.09i)15-s + 0.250·16-s + 0.192·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.823279 + 0.651919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.823279 + 0.651919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.26 - 1.18i)T \) |
| 7 | \( 1 + (-2.52 + 0.792i)T \) |
| 13 | \( 1 + (-3.46 + i)T \) |
good | 5 | \( 1 - 3.37iT - 5T^{2} \) |
| 11 | \( 1 - 5.74T + 11T^{2} \) |
| 17 | \( 1 - 0.792T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 0.147iT - 23T^{2} \) |
| 29 | \( 1 + 2.37iT - 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 - 8.36iT - 37T^{2} \) |
| 41 | \( 1 + 8.37iT - 41T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 2.74iT - 59T^{2} \) |
| 61 | \( 1 - 7.74iT - 61T^{2} \) |
| 67 | \( 1 + 5.98iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 - 6.74iT - 83T^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + 9.15T + 97T^{2} \) |
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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
−11.00030900134179790700218404393, −10.25254392724090404748680118157, −9.435235756775393684502882567371, −8.435544694119958868753722769059, −7.22311082880176453688814512406, −6.55565438621278119700156698161, −5.70503695722635526173934667176, −4.15574509224355772173576616921, −3.27329465912881978846715112476, −1.37665565081363843645827531542,
1.11004299824629714803495968392, 1.66495823251427008282235398559, 4.07801534114046681141860787923, 5.19800897871101185171250105243, 6.02195554123601111082938107354, 7.08877105423016673498236459616, 8.126391221806260117895144730119, 8.798731809368889132936541452519, 9.421751424972474743613055596006, 10.84739055767760673578614552102