L(s) = 1 | − 2.46·2-s + 5.32·3-s − 1.92·4-s − 15.0·5-s − 13.1·6-s + 24.4·8-s + 1.36·9-s + 37.1·10-s + 53.1·11-s − 10.2·12-s + 52.6·13-s − 80.2·15-s − 44.8·16-s − 118.·17-s − 3.36·18-s − 43.3·19-s + 29.0·20-s − 130.·22-s − 107.·23-s + 130.·24-s + 102.·25-s − 129.·26-s − 136.·27-s + 242.·29-s + 197.·30-s + 171.·31-s − 85.0·32-s + ⋯ |
L(s) = 1 | − 0.871·2-s + 1.02·3-s − 0.240·4-s − 1.34·5-s − 0.893·6-s + 1.08·8-s + 0.0505·9-s + 1.17·10-s + 1.45·11-s − 0.246·12-s + 1.12·13-s − 1.38·15-s − 0.701·16-s − 1.68·17-s − 0.0440·18-s − 0.523·19-s + 0.324·20-s − 1.26·22-s − 0.977·23-s + 1.10·24-s + 0.818·25-s − 0.978·26-s − 0.973·27-s + 1.55·29-s + 1.20·30-s + 0.995·31-s − 0.470·32-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 | \( 1 + 2.46T + 8T^{2} \) |
| 3 | \( 1 - 5.32T + 27T^{2} \) |
| 5 | \( 1 + 15.0T + 125T^{2} \) |
| 11 | \( 1 - 53.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 171.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 33.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 192.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 328.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 44.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 297.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 24.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 269.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 736.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 89.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 162.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 274.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 505.T + 9.12e5T^{2} \) |
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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
−8.281144312697519555858164912534, −8.020087873365597754672047112328, −6.87793539395674129887305832188, −6.29151569947962429841758231079, −4.54395620131112271770284782028, −4.10855783774000882313297286934, −3.44171717765928446656222245417, −2.16122772414457950455929061189, −1.05014811155715435963632048488, 0,
1.05014811155715435963632048488, 2.16122772414457950455929061189, 3.44171717765928446656222245417, 4.10855783774000882313297286934, 4.54395620131112271770284782028, 6.29151569947962429841758231079, 6.87793539395674129887305832188, 8.020087873365597754672047112328, 8.281144312697519555858164912534