| L(s) = 1 | − 3.33·3-s − 3.73·5-s − 3.52·7-s + 8.11·9-s − 3.60·11-s + 3.84·13-s + 12.4·15-s − 17-s + 8.37·19-s + 11.7·21-s − 3.05·23-s + 8.93·25-s − 17.0·27-s + 4.19·29-s + 7.55·31-s + 12.0·33-s + 13.1·35-s + 5.73·37-s − 12.8·39-s − 3.13·41-s − 4.16·43-s − 30.2·45-s − 8.52·47-s + 5.44·49-s + 3.33·51-s − 5.85·53-s + 13.4·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 1.66·5-s − 1.33·7-s + 2.70·9-s − 1.08·11-s + 1.06·13-s + 3.21·15-s − 0.242·17-s + 1.92·19-s + 2.56·21-s − 0.637·23-s + 1.78·25-s − 3.28·27-s + 0.779·29-s + 1.35·31-s + 2.08·33-s + 2.22·35-s + 0.943·37-s − 2.05·39-s − 0.489·41-s − 0.634·43-s − 4.51·45-s − 1.24·47-s + 0.777·49-s + 0.466·51-s − 0.803·53-s + 1.81·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3002654054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3002654054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 3.33T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 - 3.84T + 13T^{2} \) |
| 19 | \( 1 - 8.37T + 19T^{2} \) |
| 23 | \( 1 + 3.05T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 + 4.16T + 43T^{2} \) |
| 47 | \( 1 + 8.52T + 47T^{2} \) |
| 53 | \( 1 + 5.85T + 53T^{2} \) |
| 61 | \( 1 + 9.21T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 - 5.45T + 73T^{2} \) |
| 79 | \( 1 - 7.10T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.73710802768651366727125357933, −6.97586648318788007225875777077, −6.39760947596336513122894962349, −5.88705246677203751166567205982, −4.96427232321205312673010384803, −4.50620688664760819197617887401, −3.56870478348585367870026042011, −3.04858692211555730343373263858, −1.16060603533575920652624547898, −0.35763419096094299432474633726,
0.35763419096094299432474633726, 1.16060603533575920652624547898, 3.04858692211555730343373263858, 3.56870478348585367870026042011, 4.50620688664760819197617887401, 4.96427232321205312673010384803, 5.88705246677203751166567205982, 6.39760947596336513122894962349, 6.97586648318788007225875777077, 7.73710802768651366727125357933
