| L(s) = 1 | − 13.8·5-s − 30.7·7-s + 43.9·11-s − 12.2·13-s + 76.0·17-s + 44.1·19-s + 78.6·23-s + 66.6·25-s − 92.7·29-s + 143.·31-s + 425.·35-s − 32.4·37-s − 335.·41-s + 498.·43-s + 281.·47-s + 599.·49-s − 628.·53-s − 607.·55-s − 504.·59-s + 371.·61-s + 169.·65-s + 162.·67-s − 433.·71-s − 629.·73-s − 1.34e3·77-s − 172.·79-s + 174.·83-s + ⋯ |
| L(s) = 1 | − 1.23·5-s − 1.65·7-s + 1.20·11-s − 0.261·13-s + 1.08·17-s + 0.533·19-s + 0.712·23-s + 0.533·25-s − 0.594·29-s + 0.828·31-s + 2.05·35-s − 0.144·37-s − 1.27·41-s + 1.76·43-s + 0.874·47-s + 1.74·49-s − 1.62·53-s − 1.49·55-s − 1.11·59-s + 0.780·61-s + 0.323·65-s + 0.296·67-s − 0.724·71-s − 1.00·73-s − 1.99·77-s − 0.245·79-s + 0.231·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 13.8T + 125T^{2} \) |
| 7 | \( 1 + 30.7T + 343T^{2} \) |
| 11 | \( 1 - 43.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 92.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 143.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 498.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 628.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 504.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 371.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 162.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 433.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 629.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 172.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 174.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 84.3T + 9.12e5T^{2} \) |
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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
−9.067515464392435347582490837328, −7.948570632919693881939243517270, −7.20484630211908729191455647781, −6.54343031532992947795716624896, −5.62478762889175009148395066092, −4.32077829888352644563294640469, −3.55152585717758664910451993059, −2.95889211816729257396840671885, −1.08781286546959303724710566383, 0,
1.08781286546959303724710566383, 2.95889211816729257396840671885, 3.55152585717758664910451993059, 4.32077829888352644563294640469, 5.62478762889175009148395066092, 6.54343031532992947795716624896, 7.20484630211908729191455647781, 7.948570632919693881939243517270, 9.067515464392435347582490837328
