L(s) = 1 | − 0.512i·3-s + 2.30·5-s + 4.59·7-s + 8.73·9-s − 13.7·11-s − 22.2i·13-s − 1.18i·15-s + 5.43·17-s + (7.35 + 17.5i)19-s − 2.35i·21-s + 24.9·23-s − 19.6·25-s − 9.08i·27-s − 34.7i·29-s − 41.2i·31-s + ⋯ |
L(s) = 1 | − 0.170i·3-s + 0.461·5-s + 0.656·7-s + 0.970·9-s − 1.25·11-s − 1.71i·13-s − 0.0787i·15-s + 0.319·17-s + (0.386 + 0.922i)19-s − 0.112i·21-s + 1.08·23-s − 0.787·25-s − 0.336i·27-s − 1.19i·29-s − 1.33i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.225724068\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.225724068\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-7.35 - 17.5i)T \) |
good | 3 | \( 1 + 0.512iT - 9T^{2} \) |
| 5 | \( 1 - 2.30T + 25T^{2} \) |
| 7 | \( 1 - 4.59T + 49T^{2} \) |
| 11 | \( 1 + 13.7T + 121T^{2} \) |
| 13 | \( 1 + 22.2iT - 169T^{2} \) |
| 17 | \( 1 - 5.43T + 289T^{2} \) |
| 23 | \( 1 - 24.9T + 529T^{2} \) |
| 29 | \( 1 + 34.7iT - 841T^{2} \) |
| 31 | \( 1 + 41.2iT - 961T^{2} \) |
| 37 | \( 1 - 20.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 27.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 107.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 1.78iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 95.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 65.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 37.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 32.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 74.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 128. iT - 9.40e3T^{2} \) |
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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
−9.781142173296095881942946289739, −8.134362337209662298952031940973, −7.981798402053730480130431440985, −7.08152400909981276779368327716, −5.73446877878606576994534871442, −5.37373746929457797073625693917, −4.23781057468996937675343467005, −3.00858838901977980644673901411, −1.94986336236992861028353851071, −0.69633376871348169151216139132,
1.31690430358357537681731919151, 2.27232995297995693654506991583, 3.56936884621608761712814779970, 4.92739880561666323783916106582, 5.01425513129998946904959108526, 6.54388731349975543459494508099, 7.16575223167420959485667753999, 8.019284417228604679211465048372, 9.085358048994838256731178301119, 9.574381608409502054370725321559