| L(s) = 1 | + 2.67·2-s − 3-s + 5.14·4-s + 0.293·5-s − 2.67·6-s − 7-s + 8.40·8-s + 9-s + 0.783·10-s + 4.16·11-s − 5.14·12-s + 6.94·13-s − 2.67·14-s − 0.293·15-s + 12.1·16-s + 5.23·17-s + 2.67·18-s + 4.95·19-s + 1.50·20-s + 21-s + 11.1·22-s + 0.160·23-s − 8.40·24-s − 4.91·25-s + 18.5·26-s − 27-s − 5.14·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.577·3-s + 2.57·4-s + 0.131·5-s − 1.09·6-s − 0.377·7-s + 2.97·8-s + 0.333·9-s + 0.247·10-s + 1.25·11-s − 1.48·12-s + 1.92·13-s − 0.714·14-s − 0.0757·15-s + 3.04·16-s + 1.26·17-s + 0.629·18-s + 1.13·19-s + 0.337·20-s + 0.218·21-s + 2.37·22-s + 0.0334·23-s − 1.71·24-s − 0.982·25-s + 3.64·26-s − 0.192·27-s − 0.972·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.808582980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.808582980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 - 0.293T + 5T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 - 6.94T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 - 0.160T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 + 7.57T + 31T^{2} \) |
| 37 | \( 1 + 4.71T + 37T^{2} \) |
| 41 | \( 1 + 9.02T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 - 7.05T + 53T^{2} \) |
| 59 | \( 1 - 9.62T + 59T^{2} \) |
| 61 | \( 1 - 9.54T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 7.60T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 + 1.17T + 89T^{2} \) |
| 97 | \( 1 - 1.02T + 97T^{2} \) |
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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
−7.24459262248529545690218461459, −6.95272344741426150830971397756, −6.05871420427305440306930790525, −5.68356944481995767043463972572, −5.28858769399850306188619097498, −4.04188699777482854688350908669, −3.69247982667584854367897406839, −3.25135260906236672787241692655, −1.80185054785576359052218429723, −1.23014835219248027336755692287,
1.23014835219248027336755692287, 1.80185054785576359052218429723, 3.25135260906236672787241692655, 3.69247982667584854367897406839, 4.04188699777482854688350908669, 5.28858769399850306188619097498, 5.68356944481995767043463972572, 6.05871420427305440306930790525, 6.95272344741426150830971397756, 7.24459262248529545690218461459
