| L(s) = 1 | − 2-s + 0.327·3-s + 4-s − 2.37·5-s − 0.327·6-s − 2.67·7-s − 8-s − 2.89·9-s + 2.37·10-s + 2.24·11-s + 0.327·12-s − 1.42·13-s + 2.67·14-s − 0.778·15-s + 16-s − 3.35·17-s + 2.89·18-s + 0.727·19-s − 2.37·20-s − 0.875·21-s − 2.24·22-s − 0.371·23-s − 0.327·24-s + 0.637·25-s + 1.42·26-s − 1.93·27-s − 2.67·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.189·3-s + 0.5·4-s − 1.06·5-s − 0.133·6-s − 1.00·7-s − 0.353·8-s − 0.964·9-s + 0.750·10-s + 0.677·11-s + 0.0946·12-s − 0.394·13-s + 0.713·14-s − 0.200·15-s + 0.250·16-s − 0.813·17-s + 0.681·18-s + 0.166·19-s − 0.530·20-s − 0.191·21-s − 0.479·22-s − 0.0775·23-s − 0.0668·24-s + 0.127·25-s + 0.278·26-s − 0.371·27-s − 0.504·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3620959833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3620959833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 0.327T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 1.42T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 0.727T + 19T^{2} \) |
| 23 | \( 1 + 0.371T + 23T^{2} \) |
| 29 | \( 1 + 6.87T + 29T^{2} \) |
| 31 | \( 1 - 0.0885T + 31T^{2} \) |
| 37 | \( 1 + 0.422T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 6.18T + 43T^{2} \) |
| 47 | \( 1 - 6.05T + 47T^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 + 1.25T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 + 4.76T + 71T^{2} \) |
| 73 | \( 1 + 9.10T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.557398291806993985305858685053, −7.72254570313251449350283623658, −7.17197241162140827175281649169, −6.38980276767740323775651263842, −5.71470944618336892040125403433, −4.51368963657833330293738598882, −3.59442706224344415828346674722, −3.04534125182724055784602790875, −1.94893598649719167734469588126, −0.35728705706323379360689136367,
0.35728705706323379360689136367, 1.94893598649719167734469588126, 3.04534125182724055784602790875, 3.59442706224344415828346674722, 4.51368963657833330293738598882, 5.71470944618336892040125403433, 6.38980276767740323775651263842, 7.17197241162140827175281649169, 7.72254570313251449350283623658, 8.557398291806993985305858685053
