| L(s) = 1 | + 3.86·2-s + 9.18·3-s − 17.0·4-s + 25·5-s + 35.5·6-s + 171.·7-s − 189.·8-s − 158.·9-s + 96.7·10-s − 121·11-s − 156.·12-s + 17.0·13-s + 664.·14-s + 229.·15-s − 189.·16-s + 841.·17-s − 614.·18-s − 361·19-s − 425.·20-s + 1.57e3·21-s − 468.·22-s − 2.88e3·23-s − 1.74e3·24-s + 625·25-s + 65.8·26-s − 3.68e3·27-s − 2.92e3·28-s + ⋯ |
| L(s) = 1 | + 0.684·2-s + 0.589·3-s − 0.531·4-s + 0.447·5-s + 0.403·6-s + 1.32·7-s − 1.04·8-s − 0.652·9-s + 0.305·10-s − 0.301·11-s − 0.313·12-s + 0.0279·13-s + 0.905·14-s + 0.263·15-s − 0.185·16-s + 0.706·17-s − 0.446·18-s − 0.229·19-s − 0.237·20-s + 0.780·21-s − 0.206·22-s − 1.13·23-s − 0.617·24-s + 0.200·25-s + 0.0191·26-s − 0.973·27-s − 0.704·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.861392830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.861392830\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 3.86T + 32T^{2} \) |
| 3 | \( 1 - 9.18T + 243T^{2} \) |
| 7 | \( 1 - 171.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 17.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 841.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.21e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.12e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.61e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.02e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.18e5T + 8.58e9T^{2} \) |
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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
−9.050876041094617144710892957349, −8.243079938780561111389690008421, −7.929425077951929177198471559501, −6.39413721173795172706963257525, −5.53733062809157134877391618991, −4.88682667260350570418274132964, −3.99132867319619656845945800327, −2.94159231622159972177155262026, −2.05802707978185215488802371733, −0.74564046927754514556724729276,
0.74564046927754514556724729276, 2.05802707978185215488802371733, 2.94159231622159972177155262026, 3.99132867319619656845945800327, 4.88682667260350570418274132964, 5.53733062809157134877391618991, 6.39413721173795172706963257525, 7.929425077951929177198471559501, 8.243079938780561111389690008421, 9.050876041094617144710892957349
