| L(s) = 1 | + 2.00·2-s + 2.01·4-s − 4.10·5-s + 1.86·7-s + 0.0375·8-s − 8.22·10-s − 1.63·11-s + 3.48·13-s + 3.73·14-s − 3.96·16-s + 0.0762·17-s − 1.54·19-s − 8.27·20-s − 3.27·22-s − 5.94·23-s + 11.8·25-s + 6.98·26-s + 3.76·28-s + 4.65·29-s − 4.20·31-s − 8.01·32-s + 0.152·34-s − 7.64·35-s + 3.70·37-s − 3.09·38-s − 0.153·40-s + 10.6·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.00·4-s − 1.83·5-s + 0.704·7-s + 0.0132·8-s − 2.59·10-s − 0.492·11-s + 0.966·13-s + 0.999·14-s − 0.990·16-s + 0.0184·17-s − 0.354·19-s − 1.85·20-s − 0.698·22-s − 1.23·23-s + 2.36·25-s + 1.37·26-s + 0.711·28-s + 0.863·29-s − 0.755·31-s − 1.41·32-s + 0.0262·34-s − 1.29·35-s + 0.609·37-s − 0.502·38-s − 0.0243·40-s + 1.66·41-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
| \(L(1)\) |
\(\approx\) |
\(2.735770144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.735770144\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.00T + 2T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 - 0.0762T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 4.20T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 - 4.22T + 47T^{2} \) |
| 53 | \( 1 - 0.248T + 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 7.06T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 4.44T + 83T^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−7.992894316513598194401887670177, −7.37423443617678991332710009670, −6.50901712340052581479065861374, −5.78489713420259780942814476365, −4.97552759516402027668223949827, −4.23205178881497210778620303474, −3.96095585294961094516972691108, −3.16812964179203246359818349527, −2.23160943075276772744947855678, −0.68674879718373094229879513072,
0.68674879718373094229879513072, 2.23160943075276772744947855678, 3.16812964179203246359818349527, 3.96095585294961094516972691108, 4.23205178881497210778620303474, 4.97552759516402027668223949827, 5.78489713420259780942814476365, 6.50901712340052581479065861374, 7.37423443617678991332710009670, 7.992894316513598194401887670177
