| L(s) = 1 | + 2.62·2-s − 0.903·3-s + 4.90·4-s + 0.337·5-s − 2.37·6-s + 7.63·8-s − 2.18·9-s + 0.888·10-s + 11-s − 4.43·12-s + 6.09·13-s − 0.305·15-s + 10.2·16-s + 3.93·17-s − 5.74·18-s − 8.10·19-s + 1.65·20-s + 2.62·22-s + 1.82·23-s − 6.90·24-s − 4.88·25-s + 16.0·26-s + 4.68·27-s − 6.61·29-s − 0.802·30-s − 6.18·31-s + 11.6·32-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.521·3-s + 2.45·4-s + 0.151·5-s − 0.969·6-s + 2.70·8-s − 0.728·9-s + 0.280·10-s + 0.301·11-s − 1.27·12-s + 1.68·13-s − 0.0788·15-s + 2.56·16-s + 0.955·17-s − 1.35·18-s − 1.86·19-s + 0.370·20-s + 0.560·22-s + 0.381·23-s − 1.40·24-s − 0.977·25-s + 3.13·26-s + 0.901·27-s − 1.22·29-s − 0.146·30-s − 1.11·31-s + 2.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.707323652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.707323652\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 3 | \( 1 + 0.903T + 3T^{2} \) |
| 5 | \( 1 - 0.337T + 5T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 8.10T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 + 0.706T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 - 6.15T + 53T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 + 4.28T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 0.337T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
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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
−11.03679796825182042586386121055, −10.65011649820789556999645400140, −8.999525033150805031790134031799, −7.84361056197669373105479882297, −6.54469627907700594954078904506, −5.99438626241647109378253677859, −5.34766761578273655294907767502, −4.08779771422972611900721023135, −3.35349437189667357625706749500, −1.87096501944512604345109591220,
1.87096501944512604345109591220, 3.35349437189667357625706749500, 4.08779771422972611900721023135, 5.34766761578273655294907767502, 5.99438626241647109378253677859, 6.54469627907700594954078904506, 7.84361056197669373105479882297, 8.999525033150805031790134031799, 10.65011649820789556999645400140, 11.03679796825182042586386121055
