L(s) = 1 | + 4.20·2-s + 3·3-s + 9.71·4-s − 11.4·5-s + 12.6·6-s − 11.2·7-s + 7.22·8-s + 9·9-s − 48.1·10-s + 25.8·11-s + 29.1·12-s + 13·13-s − 47.3·14-s − 34.2·15-s − 47.3·16-s − 20.3·17-s + 37.8·18-s + 154.·19-s − 111.·20-s − 33.7·21-s + 108.·22-s − 180.·23-s + 21.6·24-s + 5.69·25-s + 54.7·26-s + 27·27-s − 109.·28-s + ⋯ |
L(s) = 1 | + 1.48·2-s + 0.577·3-s + 1.21·4-s − 1.02·5-s + 0.859·6-s − 0.607·7-s + 0.319·8-s + 0.333·9-s − 1.52·10-s + 0.709·11-s + 0.701·12-s + 0.277·13-s − 0.904·14-s − 0.590·15-s − 0.739·16-s − 0.290·17-s + 0.496·18-s + 1.86·19-s − 1.24·20-s − 0.350·21-s + 1.05·22-s − 1.63·23-s + 0.184·24-s + 0.0455·25-s + 0.412·26-s + 0.192·27-s − 0.738·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.527629675\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.527629675\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
good | 2 | \( 1 - 4.20T + 8T^{2} \) |
| 5 | \( 1 + 11.4T + 125T^{2} \) |
| 7 | \( 1 + 11.2T + 343T^{2} \) |
| 11 | \( 1 - 25.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 266.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 115.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 391.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 467.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 79.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 873.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 187.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 609.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 852.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 331.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 435.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 259.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−15.61092336644773112190771981736, −14.35050901270024830458836863717, −13.57424298072454794785598028302, −12.28893091620450122361463384680, −11.52880811330999102287071951704, −9.502694959866380525191190996779, −7.76775780652918002097914124192, −6.23769169350368111799495865864, −4.30788066429207028148735388895, −3.22991637083297119472160624174,
3.22991637083297119472160624174, 4.30788066429207028148735388895, 6.23769169350368111799495865864, 7.76775780652918002097914124192, 9.502694959866380525191190996779, 11.52880811330999102287071951704, 12.28893091620450122361463384680, 13.57424298072454794785598028302, 14.35050901270024830458836863717, 15.61092336644773112190771981736