| L(s) = 1 | − 0.298·2-s − 1.91·4-s − 2.54·5-s − 7-s + 1.16·8-s + 0.759·10-s − 6.15·11-s + 4.68·13-s + 0.298·14-s + 3.47·16-s + 4.41·17-s − 5.03·19-s + 4.86·20-s + 1.83·22-s − 9.11·23-s + 1.48·25-s − 1.39·26-s + 1.91·28-s + 4.28·29-s + 0.0487·31-s − 3.36·32-s − 1.31·34-s + 2.54·35-s + 3.02·37-s + 1.50·38-s − 2.96·40-s + 3.31·41-s + ⋯ |
| L(s) = 1 | − 0.210·2-s − 0.955·4-s − 1.13·5-s − 0.377·7-s + 0.412·8-s + 0.240·10-s − 1.85·11-s + 1.29·13-s + 0.0796·14-s + 0.868·16-s + 1.06·17-s − 1.15·19-s + 1.08·20-s + 0.391·22-s − 1.90·23-s + 0.297·25-s − 0.273·26-s + 0.361·28-s + 0.796·29-s + 0.00875·31-s − 0.595·32-s − 0.225·34-s + 0.430·35-s + 0.496·37-s + 0.243·38-s − 0.469·40-s + 0.518·41-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 | \( 1 + 0.298T + 2T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 11 | \( 1 + 6.15T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 23 | \( 1 + 9.11T + 23T^{2} \) |
| 29 | \( 1 - 4.28T + 29T^{2} \) |
| 31 | \( 1 - 0.0487T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 - 1.89T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 6.07T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 5.59T + 67T^{2} \) |
| 71 | \( 1 + 7.54T + 71T^{2} \) |
| 73 | \( 1 - 5.40T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 0.133T + 97T^{2} \) |
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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
−7.902761449422365607741378312908, −6.96757071000007201337236990568, −5.84641744069337973844135865472, −5.54535024510180460069908897878, −4.40083799147996761051749450467, −4.01467342180095954359450986086, −3.30305309177115694600940191701, −2.30494108515338821678894841403, −0.845250678940197524652540923616, 0,
0.845250678940197524652540923616, 2.30494108515338821678894841403, 3.30305309177115694600940191701, 4.01467342180095954359450986086, 4.40083799147996761051749450467, 5.54535024510180460069908897878, 5.84641744069337973844135865472, 6.96757071000007201337236990568, 7.902761449422365607741378312908
