| L(s) = 1 | + 2.33·2-s + 3.44·4-s + 1.04·5-s − 0.449·7-s + 3.38·8-s + 2.44·10-s + 1.28·11-s + 2·13-s − 1.04·14-s + 1.00·16-s − 2.09·17-s + 19-s + 3.61·20-s + 3·22-s − 3.61·23-s − 3.89·25-s + 4.66·26-s − 1.55·28-s + 3.38·29-s + 2.55·31-s − 4.43·32-s − 4.89·34-s − 0.471·35-s − 0.449·37-s + 2.33·38-s + 3.55·40-s − 11.6·41-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.72·4-s + 0.469·5-s − 0.169·7-s + 1.19·8-s + 0.774·10-s + 0.387·11-s + 0.554·13-s − 0.280·14-s + 0.250·16-s − 0.508·17-s + 0.229·19-s + 0.809·20-s + 0.639·22-s − 0.754·23-s − 0.779·25-s + 0.915·26-s − 0.293·28-s + 0.628·29-s + 0.458·31-s − 0.783·32-s − 0.840·34-s − 0.0797·35-s − 0.0738·37-s + 0.378·38-s + 0.561·40-s − 1.82·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.650979057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.650979057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 + 0.449T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + 0.449T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 9.34T + 43T^{2} \) |
| 47 | \( 1 + 2.33T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 6.34T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 6.34T + 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 4.43T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−11.26382894414254168207456920841, −10.22007719512078749528724353080, −9.217387414419240947020779992156, −8.018071999639162732776649321580, −6.67433713578947633230090986578, −6.17060888652275342443750620631, −5.18194303008278238446800778268, −4.18780280223572359331037781622, −3.23436303304062516804291561461, −1.95408379278428025572965956194,
1.95408379278428025572965956194, 3.23436303304062516804291561461, 4.18780280223572359331037781622, 5.18194303008278238446800778268, 6.17060888652275342443750620631, 6.67433713578947633230090986578, 8.018071999639162732776649321580, 9.217387414419240947020779992156, 10.22007719512078749528724353080, 11.26382894414254168207456920841
