L(s) = 1 | + (−0.177 − 1.99i)2-s + 1.73i·3-s + (−3.93 + 0.707i)4-s + (3.45 − 0.307i)6-s − 1.19i·7-s + (2.10 + 7.71i)8-s − 2.99·9-s − 8.22i·11-s + (−1.22 − 6.81i)12-s + 11.1·13-s + (−2.38 + 0.212i)14-s + (14.9 − 5.57i)16-s + 20.9·17-s + (0.533 + 5.97i)18-s − 27.9i·19-s + ⋯ |
L(s) = 1 | + (−0.0888 − 0.996i)2-s + 0.577i·3-s + (−0.984 + 0.176i)4-s + (0.575 − 0.0512i)6-s − 0.170i·7-s + (0.263 + 0.964i)8-s − 0.333·9-s − 0.747i·11-s + (−0.102 − 0.568i)12-s + 0.860·13-s + (−0.170 + 0.0151i)14-s + (0.937 − 0.348i)16-s + 1.23·17-s + (0.0296 + 0.332i)18-s − 1.47i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06777 - 0.892895i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06777 - 0.892895i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.177 + 1.99i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.19iT - 49T^{2} \) |
| 11 | \( 1 + 8.22iT - 121T^{2} \) |
| 13 | \( 1 - 11.1T + 169T^{2} \) |
| 17 | \( 1 - 20.9T + 289T^{2} \) |
| 19 | \( 1 + 27.9iT - 361T^{2} \) |
| 23 | \( 1 - 9.48iT - 529T^{2} \) |
| 29 | \( 1 - 40.4T + 841T^{2} \) |
| 31 | \( 1 + 55.3iT - 961T^{2} \) |
| 37 | \( 1 - 50.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 73.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 19.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 18.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 57.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 60.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 9.68iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 68.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 84.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 93.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 62.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 91.3T + 9.40e3T^{2} \) |
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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
−11.29181673632783542713109119756, −10.45835345484521713573056863447, −9.604421618418289348526741806020, −8.729160849708474418967303412585, −7.83166289599705050398112387503, −6.09773145054618278981613160729, −4.94991419242877949969150510516, −3.80813336300188900308219360497, −2.80970547938812932508564302527, −0.859282072966255892317044452875,
1.32111593870796685139384114551, 3.47315332986363849169232329441, 4.91849961889336348036859847115, 6.00139273943919923642401969308, 6.81372520817343957494096154662, 7.920049646192105221826837243699, 8.496408211446446287942028589851, 9.732465556366801051816465398895, 10.54319059832735605349276521675, 12.17466324235716359283616423460