| L(s) = 1 | − 1.76·3-s − 5-s + 4.62·7-s + 0.103·9-s + 5.52·11-s + 5.49·13-s + 1.76·15-s − 6.62·17-s + 19-s − 8.14·21-s − 4.14·23-s + 25-s + 5.10·27-s − 7.87·29-s − 1.25·31-s − 9.72·33-s − 4.62·35-s − 0.387·37-s − 9.67·39-s + 6.77·41-s + 10.9·43-s − 0.103·45-s − 1.72·47-s + 14.4·49-s + 11.6·51-s + 1.49·53-s − 5.52·55-s + ⋯ |
| L(s) = 1 | − 1.01·3-s − 0.447·5-s + 1.74·7-s + 0.0343·9-s + 1.66·11-s + 1.52·13-s + 0.454·15-s − 1.60·17-s + 0.229·19-s − 1.77·21-s − 0.865·23-s + 0.200·25-s + 0.982·27-s − 1.46·29-s − 0.224·31-s − 1.69·33-s − 0.781·35-s − 0.0637·37-s − 1.54·39-s + 1.05·41-s + 1.67·43-s − 0.0153·45-s − 0.252·47-s + 2.05·49-s + 1.63·51-s + 0.204·53-s − 0.744·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.444605125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444605125\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 7 | \( 1 - 4.62T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 + 6.62T + 17T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 0.387T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 + 0.626T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.83T + 73T^{2} \) |
| 79 | \( 1 - 2.98T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 - 4.27T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−9.218297968457847557802224730657, −8.700009052305009051099604097653, −7.927434188684470920826000445948, −6.91267215541552086867782300793, −6.12944158579867901226943604447, −5.42674283011219229946173644213, −4.30015919988280673667498432821, −3.93547274482722403538255479999, −1.98188312223055493445124120904, −0.963802058085393332622397234047,
0.963802058085393332622397234047, 1.98188312223055493445124120904, 3.93547274482722403538255479999, 4.30015919988280673667498432821, 5.42674283011219229946173644213, 6.12944158579867901226943604447, 6.91267215541552086867782300793, 7.927434188684470920826000445948, 8.700009052305009051099604097653, 9.218297968457847557802224730657
