| L(s) = 1 | − 0.571·2-s − 1.67·4-s − 3.31·7-s + 2.10·8-s + 0.778·11-s − 3.69·13-s + 1.89·14-s + 2.14·16-s − 4.20·17-s + 5.46·19-s − 0.445·22-s + 4.59·23-s + 2.11·26-s + 5.54·28-s − 2.33·29-s + 5.95·31-s − 5.42·32-s + 2.40·34-s + 37-s − 3.12·38-s − 8.98·41-s − 8.17·43-s − 1.30·44-s − 2.62·46-s − 2.91·47-s + 3.97·49-s + 6.17·52-s + ⋯ |
| L(s) = 1 | − 0.404·2-s − 0.836·4-s − 1.25·7-s + 0.742·8-s + 0.234·11-s − 1.02·13-s + 0.506·14-s + 0.536·16-s − 1.01·17-s + 1.25·19-s − 0.0948·22-s + 0.957·23-s + 0.414·26-s + 1.04·28-s − 0.433·29-s + 1.06·31-s − 0.959·32-s + 0.412·34-s + 0.164·37-s − 0.507·38-s − 1.40·41-s − 1.24·43-s − 0.196·44-s − 0.387·46-s − 0.425·47-s + 0.568·49-s + 0.856·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 0.571T + 2T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 - 0.778T + 11T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 4.59T + 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 8.17T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 - 9.44T + 67T^{2} \) |
| 71 | \( 1 - 1.78T + 71T^{2} \) |
| 73 | \( 1 + 5.52T + 73T^{2} \) |
| 79 | \( 1 - 1.93T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 9.74T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−7.39819306730107886204438366782, −6.87598923884271795777056093856, −6.22979855166862707768871025786, −5.08092884082397754836426382720, −4.89112122240340313699904011204, −3.74326227480510940471821043535, −3.21950218904674249457161738512, −2.21704460708355021941059525885, −0.935786472358429757929254438034, 0,
0.935786472358429757929254438034, 2.21704460708355021941059525885, 3.21950218904674249457161738512, 3.74326227480510940471821043535, 4.89112122240340313699904011204, 5.08092884082397754836426382720, 6.22979855166862707768871025786, 6.87598923884271795777056093856, 7.39819306730107886204438366782
