| L(s) = 1 | − 1.82·2-s + 0.0328·3-s + 1.32·4-s − 0.181·5-s − 0.0599·6-s + 4.41·7-s + 1.22·8-s − 2.99·9-s + 0.331·10-s − 0.871·11-s + 0.0436·12-s + 3.03·13-s − 8.06·14-s − 0.00596·15-s − 4.89·16-s + 6.34·17-s + 5.47·18-s − 0.715·19-s − 0.241·20-s + 0.145·21-s + 1.58·22-s + 2.85·23-s + 0.0401·24-s − 4.96·25-s − 5.53·26-s − 0.196·27-s + 5.87·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.0189·3-s + 0.664·4-s − 0.0812·5-s − 0.0244·6-s + 1.67·7-s + 0.432·8-s − 0.999·9-s + 0.104·10-s − 0.262·11-s + 0.0126·12-s + 0.841·13-s − 2.15·14-s − 0.00154·15-s − 1.22·16-s + 1.53·17-s + 1.28·18-s − 0.164·19-s − 0.0540·20-s + 0.0316·21-s + 0.338·22-s + 0.594·23-s + 0.00820·24-s − 0.993·25-s − 1.08·26-s − 0.0379·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.164527705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.164527705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 - 0.0328T + 3T^{2} \) |
| 5 | \( 1 + 0.181T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + 0.871T + 11T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 19 | \( 1 + 0.715T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 - 7.22T + 31T^{2} \) |
| 41 | \( 1 + 9.57T + 41T^{2} \) |
| 43 | \( 1 + 7.97T + 43T^{2} \) |
| 47 | \( 1 - 7.45T + 47T^{2} \) |
| 53 | \( 1 - 2.08T + 53T^{2} \) |
| 59 | \( 1 - 9.13T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 + 7.64T + 67T^{2} \) |
| 71 | \( 1 - 0.273T + 71T^{2} \) |
| 73 | \( 1 - 3.37T + 73T^{2} \) |
| 79 | \( 1 - 2.85T + 79T^{2} \) |
| 83 | \( 1 + 1.37T + 83T^{2} \) |
| 89 | \( 1 + 1.61T + 89T^{2} \) |
| 97 | \( 1 - 6.51T + 97T^{2} \) |
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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
−8.372037300783374523116389522086, −8.078379289960870665394126645816, −7.39983105271607218952851399642, −6.37886097224468545071709970967, −5.40004385425401852624406975090, −4.86263938644974701604262606649, −3.79283612257634570579865542694, −2.63924953966276910141153097220, −1.58794350300465029236664368680, −0.819179662463666870092227944912,
0.819179662463666870092227944912, 1.58794350300465029236664368680, 2.63924953966276910141153097220, 3.79283612257634570579865542694, 4.86263938644974701604262606649, 5.40004385425401852624406975090, 6.37886097224468545071709970967, 7.39983105271607218952851399642, 8.078379289960870665394126645816, 8.372037300783374523116389522086
