| L(s) = 1 | − 2.44·2-s − 2.32·3-s + 3.95·4-s − 3.72·5-s + 5.66·6-s − 3.99·7-s − 4.77·8-s + 2.38·9-s + 9.09·10-s − 11-s − 9.17·12-s + 1.34·13-s + 9.75·14-s + 8.65·15-s + 3.73·16-s − 17-s − 5.81·18-s − 2.39·19-s − 14.7·20-s + 9.27·21-s + 2.44·22-s + 6.85·23-s + 11.0·24-s + 8.89·25-s − 3.29·26-s + 1.43·27-s − 15.8·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 1.33·3-s + 1.97·4-s − 1.66·5-s + 2.31·6-s − 1.51·7-s − 1.68·8-s + 0.794·9-s + 2.87·10-s − 0.301·11-s − 2.64·12-s + 0.374·13-s + 2.60·14-s + 2.23·15-s + 0.932·16-s − 0.242·17-s − 1.37·18-s − 0.549·19-s − 3.29·20-s + 2.02·21-s + 0.520·22-s + 1.43·23-s + 2.25·24-s + 1.77·25-s − 0.646·26-s + 0.275·27-s − 2.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2381310153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2381310153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.32T + 3T^{2} \) |
| 5 | \( 1 + 3.72T + 5T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 13 | \( 1 - 1.34T + 13T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 - 2.30T + 61T^{2} \) |
| 67 | \( 1 - 5.30T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 3.87T + 79T^{2} \) |
| 83 | \( 1 + 3.77T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−7.82273506682017037921657779210, −7.19718302113197795446384233122, −6.49918087064292694559090490654, −6.37047590408468381446517430721, −5.13357389598244229796596105476, −4.28720244460646670825652328081, −3.33338279247603895312070496504, −2.58285520658876431023286400151, −0.831421398111931443055745818740, −0.51358528152165695596457088751,
0.51358528152165695596457088751, 0.831421398111931443055745818740, 2.58285520658876431023286400151, 3.33338279247603895312070496504, 4.28720244460646670825652328081, 5.13357389598244229796596105476, 6.37047590408468381446517430721, 6.49918087064292694559090490654, 7.19718302113197795446384233122, 7.82273506682017037921657779210
