| L(s) = 1 | − 23.6·7-s − 9.73·11-s − 71.8·13-s − 135.·17-s − 49.1·19-s − 191.·23-s − 45.6·29-s + 32.1·31-s + 218.·37-s + 394.·41-s + 396.·43-s + 33.3·47-s + 218.·49-s − 150.·53-s − 396.·59-s − 505.·61-s − 552.·67-s + 756.·71-s − 579.·73-s + 230.·77-s + 18.6·79-s + 1.31e3·83-s + 541.·89-s + 1.70e3·91-s − 16.7·97-s − 705.·101-s − 442.·103-s + ⋯ |
| L(s) = 1 | − 1.27·7-s − 0.266·11-s − 1.53·13-s − 1.94·17-s − 0.593·19-s − 1.73·23-s − 0.291·29-s + 0.186·31-s + 0.970·37-s + 1.50·41-s + 1.40·43-s + 0.103·47-s + 0.637·49-s − 0.390·53-s − 0.874·59-s − 1.06·61-s − 1.00·67-s + 1.26·71-s − 0.929·73-s + 0.341·77-s + 0.0265·79-s + 1.73·83-s + 0.644·89-s + 1.96·91-s − 0.0175·97-s − 0.695·101-s − 0.422·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4185430240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4185430240\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23.6T + 343T^{2} \) |
| 11 | \( 1 + 9.73T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 135.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 45.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 32.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 394.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 396.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 33.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 396.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 505.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 552.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 756.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 579.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 18.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.31e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 541.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 16.7T + 9.12e5T^{2} \) |
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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
−9.259088287056673813956396352877, −8.030384168349645005089889583361, −7.32670155656850826873710386169, −6.42771361631794692541281327529, −5.94006509704424967882974233030, −4.63759746205745139674933525377, −4.05051272615441295624648813620, −2.72457188590316290769984099101, −2.18059895173987945960610441213, −0.27914176987962886349668708815,
0.27914176987962886349668708815, 2.18059895173987945960610441213, 2.72457188590316290769984099101, 4.05051272615441295624648813620, 4.63759746205745139674933525377, 5.94006509704424967882974233030, 6.42771361631794692541281327529, 7.32670155656850826873710386169, 8.030384168349645005089889583361, 9.259088287056673813956396352877
