| L(s) = 1 | − 1.27·2-s − 0.378·4-s + 2.08·5-s + 7-s + 3.02·8-s − 2.65·10-s + 2.11·11-s − 3.46·13-s − 1.27·14-s − 3.10·16-s − 5.03·17-s + 0.224·19-s − 0.789·20-s − 2.68·22-s + 7.75·23-s − 0.649·25-s + 4.41·26-s − 0.378·28-s − 5.57·29-s − 0.678·31-s − 2.10·32-s + 6.40·34-s + 2.08·35-s + 10.9·37-s − 0.285·38-s + 6.31·40-s − 1.33·41-s + ⋯ |
| L(s) = 1 | − 0.900·2-s − 0.189·4-s + 0.932·5-s + 0.377·7-s + 1.07·8-s − 0.839·10-s + 0.636·11-s − 0.961·13-s − 0.340·14-s − 0.775·16-s − 1.22·17-s + 0.0514·19-s − 0.176·20-s − 0.573·22-s + 1.61·23-s − 0.129·25-s + 0.865·26-s − 0.0715·28-s − 1.03·29-s − 0.121·31-s − 0.372·32-s + 1.09·34-s + 0.352·35-s + 1.79·37-s − 0.0463·38-s + 0.998·40-s − 0.208·41-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 | \( 1 + 1.27T + 2T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 5.03T + 17T^{2} \) |
| 19 | \( 1 - 0.224T + 19T^{2} \) |
| 23 | \( 1 - 7.75T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 + 0.678T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 1.33T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + 0.612T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 - 0.604T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 7.12T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 - 5.41T + 79T^{2} \) |
| 83 | \( 1 + 5.01T + 83T^{2} \) |
| 89 | \( 1 + 8.62T + 89T^{2} \) |
| 97 | \( 1 - 6.43T + 97T^{2} \) |
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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
−7.57917171513614152772447347951, −6.93185740028706589358184296878, −6.26777603792509147953792528764, −5.30442201848888592656105360103, −4.74950625709710896797072467422, −4.06710667206238230247728499201, −2.82187072967682952640315008712, −1.94745489021789217979004090629, −1.26524132231775778186577920827, 0,
1.26524132231775778186577920827, 1.94745489021789217979004090629, 2.82187072967682952640315008712, 4.06710667206238230247728499201, 4.74950625709710896797072467422, 5.30442201848888592656105360103, 6.26777603792509147953792528764, 6.93185740028706589358184296878, 7.57917171513614152772447347951
