| L(s) = 1 | − 1.89·2-s + 1.60·4-s − 4.06·5-s + 3.06·7-s + 0.749·8-s + 7.71·10-s − 1.26·11-s − 2.45·13-s − 5.81·14-s − 4.63·16-s + 4.97·17-s − 2.32·19-s − 6.51·20-s + 2.39·22-s + 4.38·23-s + 11.4·25-s + 4.66·26-s + 4.91·28-s + 3.25·29-s − 31-s + 7.29·32-s − 9.44·34-s − 12.4·35-s − 6.46·37-s + 4.41·38-s − 3.04·40-s − 10.7·41-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.802·4-s − 1.81·5-s + 1.15·7-s + 0.265·8-s + 2.43·10-s − 0.381·11-s − 0.681·13-s − 1.55·14-s − 1.15·16-s + 1.20·17-s − 0.533·19-s − 1.45·20-s + 0.511·22-s + 0.914·23-s + 2.29·25-s + 0.915·26-s + 0.928·28-s + 0.604·29-s − 0.179·31-s + 1.29·32-s − 1.62·34-s − 2.10·35-s − 1.06·37-s + 0.716·38-s − 0.481·40-s − 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.89T + 2T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 7 | \( 1 - 3.06T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 2.45T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 9.75T + 53T^{2} \) |
| 59 | \( 1 + 5.53T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 + 2.26T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−9.739310905813203376678207451694, −8.586062426699766538116227087754, −8.195416258529565667392137952497, −7.55736558464770981415136538426, −6.96511233255147954789094481207, −5.07646986881289861029131567895, −4.43192232672828851201893621851, −3.10939754213122391053477978382, −1.41776868483765792216077011057, 0,
1.41776868483765792216077011057, 3.10939754213122391053477978382, 4.43192232672828851201893621851, 5.07646986881289861029131567895, 6.96511233255147954789094481207, 7.55736558464770981415136538426, 8.195416258529565667392137952497, 8.586062426699766538116227087754, 9.739310905813203376678207451694
