| L(s) = 1 | + 0.243·2-s − 1.94·4-s + 0.214·5-s − 3.34·7-s − 0.960·8-s + 0.0523·10-s − 0.762·11-s − 0.823·13-s − 0.815·14-s + 3.64·16-s − 6.98·17-s − 3.85·19-s − 0.416·20-s − 0.185·22-s − 6.58·23-s − 4.95·25-s − 0.200·26-s + 6.49·28-s + 6.80·29-s − 3.89·31-s + 2.80·32-s − 1.70·34-s − 0.718·35-s − 0.891·37-s − 0.940·38-s − 0.206·40-s + 4.22·41-s + ⋯ |
| L(s) = 1 | + 0.172·2-s − 0.970·4-s + 0.0959·5-s − 1.26·7-s − 0.339·8-s + 0.0165·10-s − 0.229·11-s − 0.228·13-s − 0.217·14-s + 0.911·16-s − 1.69·17-s − 0.885·19-s − 0.0931·20-s − 0.0396·22-s − 1.37·23-s − 0.990·25-s − 0.0393·26-s + 1.22·28-s + 1.26·29-s − 0.699·31-s + 0.496·32-s − 0.292·34-s − 0.121·35-s − 0.146·37-s − 0.152·38-s − 0.0325·40-s + 0.659·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3589234933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3589234933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 0.243T + 2T^{2} \) |
| 5 | \( 1 - 0.214T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 0.762T + 11T^{2} \) |
| 13 | \( 1 + 0.823T + 13T^{2} \) |
| 17 | \( 1 + 6.98T + 17T^{2} \) |
| 19 | \( 1 + 3.85T + 19T^{2} \) |
| 23 | \( 1 + 6.58T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 3.89T + 31T^{2} \) |
| 37 | \( 1 + 0.891T + 37T^{2} \) |
| 41 | \( 1 - 4.22T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 + 3.50T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7.52T + 67T^{2} \) |
| 71 | \( 1 - 2.52T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 + 9.16T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 + 2.46T + 97T^{2} \) |
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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
−8.259883545570337027529550971243, −7.35824723137373470590308716824, −6.33528646328728881253189385175, −6.17840128631282172723506964368, −5.12758668236485876725106898700, −4.31482312750921794959986810966, −3.83544523852843755240756789927, −2.86539575021957742433084839844, −1.97880927360598741497278859237, −0.28971003483688789333521712113,
0.28971003483688789333521712113, 1.97880927360598741497278859237, 2.86539575021957742433084839844, 3.83544523852843755240756789927, 4.31482312750921794959986810966, 5.12758668236485876725106898700, 6.17840128631282172723506964368, 6.33528646328728881253189385175, 7.35824723137373470590308716824, 8.259883545570337027529550971243
