| L(s) = 1 | − 0.146·2-s − 1.97·4-s + 2.93·5-s − 7-s + 0.582·8-s − 0.429·10-s + 1.72·11-s − 1.53·13-s + 0.146·14-s + 3.87·16-s + 1.53·17-s + 6.29·19-s − 5.80·20-s − 0.252·22-s + 0.209·23-s + 3.60·25-s + 0.224·26-s + 1.97·28-s − 9.36·29-s − 5.71·31-s − 1.73·32-s − 0.224·34-s − 2.93·35-s − 9.72·37-s − 0.921·38-s + 1.70·40-s − 8.99·41-s + ⋯ |
| L(s) = 1 | − 0.103·2-s − 0.989·4-s + 1.31·5-s − 0.377·7-s + 0.205·8-s − 0.135·10-s + 0.520·11-s − 0.424·13-s + 0.0391·14-s + 0.967·16-s + 0.371·17-s + 1.44·19-s − 1.29·20-s − 0.0539·22-s + 0.0436·23-s + 0.720·25-s + 0.0439·26-s + 0.373·28-s − 1.73·29-s − 1.02·31-s − 0.306·32-s − 0.0384·34-s − 0.495·35-s − 1.59·37-s − 0.149·38-s + 0.270·40-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.146T + 2T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 - 0.209T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 + 5.71T + 31T^{2} \) |
| 37 | \( 1 + 9.72T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 0.690T + 47T^{2} \) |
| 53 | \( 1 + 6.60T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 + 7.54T + 71T^{2} \) |
| 73 | \( 1 - 0.853T + 73T^{2} \) |
| 79 | \( 1 - 1.08T + 79T^{2} \) |
| 83 | \( 1 + 6.30T + 83T^{2} \) |
| 89 | \( 1 - 0.111T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.39255563549573466584027724961, −6.86381872881511725280650013259, −5.91042308424513302739442363060, −5.30528964808173031680037777169, −5.01401081484566343215137093015, −3.67235559074242900880954248273, −3.35473774564124091702787724500, −2.01192705644095205267528885149, −1.33920919979949778972273399981, 0,
1.33920919979949778972273399981, 2.01192705644095205267528885149, 3.35473774564124091702787724500, 3.67235559074242900880954248273, 5.01401081484566343215137093015, 5.30528964808173031680037777169, 5.91042308424513302739442363060, 6.86381872881511725280650013259, 7.39255563549573466584027724961
