| L(s) = 1 | − 2.18·2-s + 2.76·4-s − 4.31·7-s − 1.67·8-s − 4.00·11-s + 4.18·13-s + 9.41·14-s − 1.87·16-s − 3.22·17-s − 7.97·19-s + 8.75·22-s + 7.08·23-s − 9.14·26-s − 11.9·28-s + 0.506·29-s − 3·31-s + 7.44·32-s + 7.04·34-s − 37-s + 17.4·38-s + 1.46·41-s + 10.8·43-s − 11.0·44-s − 15.4·46-s − 3.28·47-s + 11.6·49-s + 11.5·52-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.38·4-s − 1.63·7-s − 0.593·8-s − 1.20·11-s + 1.16·13-s + 2.51·14-s − 0.468·16-s − 0.782·17-s − 1.82·19-s + 1.86·22-s + 1.47·23-s − 1.79·26-s − 2.25·28-s + 0.0940·29-s − 0.538·31-s + 1.31·32-s + 1.20·34-s − 0.164·37-s + 2.82·38-s + 0.228·41-s + 1.66·43-s − 1.67·44-s − 2.28·46-s − 0.479·47-s + 1.65·49-s + 1.60·52-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 | \( 1 + 2.18T + 2T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 + 4.00T + 11T^{2} \) |
| 13 | \( 1 - 4.18T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 + 7.97T + 19T^{2} \) |
| 23 | \( 1 - 7.08T + 23T^{2} \) |
| 29 | \( 1 - 0.506T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 41 | \( 1 - 1.46T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 3.32T + 59T^{2} \) |
| 61 | \( 1 - 2.88T + 61T^{2} \) |
| 67 | \( 1 - 0.164T + 67T^{2} \) |
| 71 | \( 1 + 0.883T + 71T^{2} \) |
| 73 | \( 1 + 6.13T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 1.12T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 + 3.46T + 97T^{2} \) |
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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
−7.47022408667479335531200109238, −6.94918906622123841190781873863, −6.34090772659150728658642640962, −5.74554682811771030588248235752, −4.57281431183508857187397892524, −3.70268877145168177661444495327, −2.74539687725721594200099266410, −2.16231400080553389119116826849, −0.833138258095476621713761162627, 0,
0.833138258095476621713761162627, 2.16231400080553389119116826849, 2.74539687725721594200099266410, 3.70268877145168177661444495327, 4.57281431183508857187397892524, 5.74554682811771030588248235752, 6.34090772659150728658642640962, 6.94918906622123841190781873863, 7.47022408667479335531200109238
