| L(s) = 1 | − 1.28·2-s − 1.89·3-s − 6.34·4-s − 17.6·5-s + 2.44·6-s + 18.4·8-s − 23.4·9-s + 22.7·10-s − 54.3·11-s + 12.0·12-s − 22.3·13-s + 33.5·15-s + 26.9·16-s − 18.8·17-s + 30.1·18-s − 142.·19-s + 112.·20-s + 69.9·22-s − 68.8·23-s − 35.0·24-s + 187.·25-s + 28.7·26-s + 95.5·27-s − 21.1·29-s − 43.1·30-s + 288.·31-s − 182.·32-s + ⋯ |
| L(s) = 1 | − 0.455·2-s − 0.364·3-s − 0.792·4-s − 1.58·5-s + 0.166·6-s + 0.816·8-s − 0.866·9-s + 0.720·10-s − 1.49·11-s + 0.289·12-s − 0.476·13-s + 0.577·15-s + 0.421·16-s − 0.269·17-s + 0.394·18-s − 1.71·19-s + 1.25·20-s + 0.678·22-s − 0.624·23-s − 0.297·24-s + 1.50·25-s + 0.217·26-s + 0.681·27-s − 0.135·29-s − 0.262·30-s + 1.66·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.28T + 8T^{2} \) |
| 3 | \( 1 + 1.89T + 27T^{2} \) |
| 5 | \( 1 + 17.6T + 125T^{2} \) |
| 11 | \( 1 + 54.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 22.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 300.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 392.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 650.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 158.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 66.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 755.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 418.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 92.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 647.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 458.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 528.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.09e3T + 9.12e5T^{2} \) |
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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
−8.220576964257091826616882831203, −7.85726884008641884013500804465, −6.85976617956946601027554448128, −5.81182354510114621793073224336, −4.79377299256444374334896297561, −4.42786597037212791515733094068, −3.36976538579578977740412594453, −2.34840643636648182047284315365, −0.54310736607978352073311068313, 0,
0.54310736607978352073311068313, 2.34840643636648182047284315365, 3.36976538579578977740412594453, 4.42786597037212791515733094068, 4.79377299256444374334896297561, 5.81182354510114621793073224336, 6.85976617956946601027554448128, 7.85726884008641884013500804465, 8.220576964257091826616882831203
