| L(s) = 1 | − 2.32·2-s + 3.41·4-s + 1.82·5-s + 7-s − 3.29·8-s − 4.25·10-s + 3.25·11-s + 0.459·13-s − 2.32·14-s + 0.837·16-s + 0.923·17-s + 1.76·19-s + 6.24·20-s − 7.56·22-s − 8.20·23-s − 1.65·25-s − 1.06·26-s + 3.41·28-s + 7.06·29-s − 8.26·31-s + 4.64·32-s − 2.14·34-s + 1.82·35-s + 2.64·37-s − 4.11·38-s − 6.02·40-s − 5.05·41-s + ⋯ |
| L(s) = 1 | − 1.64·2-s + 1.70·4-s + 0.817·5-s + 0.377·7-s − 1.16·8-s − 1.34·10-s + 0.980·11-s + 0.127·13-s − 0.621·14-s + 0.209·16-s + 0.224·17-s + 0.405·19-s + 1.39·20-s − 1.61·22-s − 1.71·23-s − 0.331·25-s − 0.209·26-s + 0.645·28-s + 1.31·29-s − 1.48·31-s + 0.820·32-s − 0.368·34-s + 0.308·35-s + 0.434·37-s − 0.667·38-s − 0.952·40-s − 0.789·41-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 | \( 1 + 2.32T + 2T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 0.459T + 13T^{2} \) |
| 17 | \( 1 - 0.923T + 17T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 + 8.20T + 23T^{2} \) |
| 29 | \( 1 - 7.06T + 29T^{2} \) |
| 31 | \( 1 + 8.26T + 31T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 - 1.18T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 + 5.89T + 61T^{2} \) |
| 67 | \( 1 + 6.38T + 67T^{2} \) |
| 71 | \( 1 + 4.59T + 71T^{2} \) |
| 73 | \( 1 + 5.10T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.57705599817435960695263469211, −7.06655680052238634315675701938, −6.19721605696957921139542925186, −5.81223258266286652178429311979, −4.69623620910717156482302877650, −3.79285745608379695508624394761, −2.69123255022537750384805821649, −1.71559079849695157673441145660, −1.38612314200964815301375324847, 0,
1.38612314200964815301375324847, 1.71559079849695157673441145660, 2.69123255022537750384805821649, 3.79285745608379695508624394761, 4.69623620910717156482302877650, 5.81223258266286652178429311979, 6.19721605696957921139542925186, 7.06655680052238634315675701938, 7.57705599817435960695263469211
