| L(s) = 1 | − 1.00·3-s + 2.49·5-s − 7-s − 1.98·9-s + 11-s + 13-s − 2.52·15-s − 4.53·17-s − 0.365·19-s + 1.00·21-s + 2.18·23-s + 1.23·25-s + 5.02·27-s + 7.70·29-s − 6.21·31-s − 1.00·33-s − 2.49·35-s − 6.41·37-s − 1.00·39-s − 0.881·41-s − 2.18·43-s − 4.94·45-s + 13.5·47-s + 49-s + 4.57·51-s − 5.98·53-s + 2.49·55-s + ⋯ |
| L(s) = 1 | − 0.582·3-s + 1.11·5-s − 0.377·7-s − 0.660·9-s + 0.301·11-s + 0.277·13-s − 0.651·15-s − 1.09·17-s − 0.0839·19-s + 0.220·21-s + 0.455·23-s + 0.247·25-s + 0.967·27-s + 1.43·29-s − 1.11·31-s − 0.175·33-s − 0.422·35-s − 1.05·37-s − 0.161·39-s − 0.137·41-s − 0.332·43-s − 0.737·45-s + 1.97·47-s + 0.142·49-s + 0.641·51-s − 0.821·53-s + 0.336·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 1.00T + 3T^{2} \) |
| 5 | \( 1 - 2.49T + 5T^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 19 | \( 1 + 0.365T + 19T^{2} \) |
| 23 | \( 1 - 2.18T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 + 6.21T + 31T^{2} \) |
| 37 | \( 1 + 6.41T + 37T^{2} \) |
| 41 | \( 1 + 0.881T + 41T^{2} \) |
| 43 | \( 1 + 2.18T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + 5.98T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 4.84T + 61T^{2} \) |
| 67 | \( 1 + 0.801T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 2.15T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−7.18823707481763752534769756140, −6.67543422337086333898482664794, −6.04467148776705843815215991050, −5.56061393060079526550361963313, −4.85445266321019468983192437644, −3.96102693417568908188611355585, −2.95035979390530377935584761874, −2.24681862179079084309703973190, −1.24853594261325381188601390313, 0,
1.24853594261325381188601390313, 2.24681862179079084309703973190, 2.95035979390530377935584761874, 3.96102693417568908188611355585, 4.85445266321019468983192437644, 5.56061393060079526550361963313, 6.04467148776705843815215991050, 6.67543422337086333898482664794, 7.18823707481763752534769756140
