| L(s) = 1 | + 1.90·5-s − 2.28·7-s + 2.68·11-s − 4.18·13-s − 17-s − 0.689·19-s + 3.39·23-s − 1.38·25-s + 5.86·29-s + 7.86·31-s − 4.34·35-s − 5.86·37-s − 2.61·41-s + 2.88·43-s + 3.80·47-s − 1.77·49-s + 0.423·53-s + 5.11·55-s + 3.57·59-s + 3.63·61-s − 7.96·65-s + 2.64·67-s + 12.4·71-s + 8.14·73-s − 6.14·77-s + 10.0·79-s − 2.57·83-s + ⋯ |
| L(s) = 1 | + 0.850·5-s − 0.864·7-s + 0.810·11-s − 1.16·13-s − 0.242·17-s − 0.158·19-s + 0.708·23-s − 0.277·25-s + 1.08·29-s + 1.41·31-s − 0.734·35-s − 0.963·37-s − 0.407·41-s + 0.440·43-s + 0.554·47-s − 0.253·49-s + 0.0581·53-s + 0.689·55-s + 0.465·59-s + 0.465·61-s − 0.987·65-s + 0.323·67-s + 1.47·71-s + 0.953·73-s − 0.700·77-s + 1.13·79-s − 0.282·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.977024796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977024796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 - 2.68T + 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 19 | \( 1 + 0.689T + 19T^{2} \) |
| 23 | \( 1 - 3.39T + 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 - 2.88T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 - 0.423T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 - 2.64T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 8.14T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 2.57T + 83T^{2} \) |
| 89 | \( 1 + 7.37T + 89T^{2} \) |
| 97 | \( 1 - 4.02T + 97T^{2} \) |
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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
−8.401830996302521599060292807745, −7.38961492541064744746868010421, −6.60306845861505100876930847200, −6.34434190361102115135175896451, −5.32274135209037474489527326729, −4.66622988943898236861887273048, −3.67527800322157614430546220648, −2.77215785680720940908655419730, −2.03330794088023983853831735391, −0.76414961271303616860381933082,
0.76414961271303616860381933082, 2.03330794088023983853831735391, 2.77215785680720940908655419730, 3.67527800322157614430546220648, 4.66622988943898236861887273048, 5.32274135209037474489527326729, 6.34434190361102115135175896451, 6.60306845861505100876930847200, 7.38961492541064744746868010421, 8.401830996302521599060292807745
