| L(s) = 1 | − 1.67·2-s − 2.99·3-s + 0.811·4-s + 2.10·5-s + 5.02·6-s + 1.99·8-s + 5.98·9-s − 3.52·10-s + 0.0709·11-s − 2.43·12-s + 7.15·13-s − 6.30·15-s − 4.96·16-s − 4.51·17-s − 10.0·18-s − 2.31·19-s + 1.70·20-s − 0.118·22-s − 4.52·23-s − 5.97·24-s − 0.572·25-s − 12.0·26-s − 8.94·27-s − 4.64·29-s + 10.5·30-s + 3.65·31-s + 4.33·32-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 1.73·3-s + 0.405·4-s + 0.940·5-s + 2.05·6-s + 0.704·8-s + 1.99·9-s − 1.11·10-s + 0.0213·11-s − 0.701·12-s + 1.98·13-s − 1.62·15-s − 1.24·16-s − 1.09·17-s − 2.36·18-s − 0.531·19-s + 0.381·20-s − 0.0253·22-s − 0.942·23-s − 1.21·24-s − 0.114·25-s − 2.35·26-s − 1.72·27-s − 0.862·29-s + 1.93·30-s + 0.657·31-s + 0.766·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 - 2.10T + 5T^{2} \) |
| 11 | \( 1 - 0.0709T + 11T^{2} \) |
| 13 | \( 1 - 7.15T + 13T^{2} \) |
| 17 | \( 1 + 4.51T + 17T^{2} \) |
| 19 | \( 1 + 2.31T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 + 4.64T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 - 3.50T + 37T^{2} \) |
| 41 | \( 1 + 0.176T + 41T^{2} \) |
| 43 | \( 1 + 8.71T + 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 + 0.948T + 53T^{2} \) |
| 59 | \( 1 - 2.64T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 3.18T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 1.12T + 73T^{2} \) |
| 79 | \( 1 + 0.707T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.756507843297995280964153702990, −7.916455755812005798696600598857, −6.84788943751184670086971017415, −6.19722117320480339424869853624, −5.82268942545319675975492926437, −4.72379164232301320912106921610, −3.96291901994390930834040953854, −1.96693385525227938903197099587, −1.21995754567916358888310142375, 0,
1.21995754567916358888310142375, 1.96693385525227938903197099587, 3.96291901994390930834040953854, 4.72379164232301320912106921610, 5.82268942545319675975492926437, 6.19722117320480339424869853624, 6.84788943751184670086971017415, 7.916455755812005798696600598857, 8.756507843297995280964153702990
