| L(s) = 1 | − 1.58·3-s − 5-s + 3.33·7-s − 0.500·9-s − 4.48·11-s + 5.14·13-s + 1.58·15-s − 2.08·17-s − 1.21·19-s − 5.27·21-s − 6.31·23-s + 25-s + 5.53·27-s − 2.70·29-s + 9.30·31-s + 7.08·33-s − 3.33·35-s − 2.68·37-s − 8.13·39-s + 10.6·41-s + 9.65·43-s + 0.500·45-s − 12.9·47-s + 4.11·49-s + 3.29·51-s − 1.27·53-s + 4.48·55-s + ⋯ |
| L(s) = 1 | − 0.912·3-s − 0.447·5-s + 1.25·7-s − 0.166·9-s − 1.35·11-s + 1.42·13-s + 0.408·15-s − 0.505·17-s − 0.279·19-s − 1.15·21-s − 1.31·23-s + 0.200·25-s + 1.06·27-s − 0.501·29-s + 1.67·31-s + 1.23·33-s − 0.563·35-s − 0.442·37-s − 1.30·39-s + 1.66·41-s + 1.47·43-s + 0.0745·45-s − 1.88·47-s + 0.587·49-s + 0.461·51-s − 0.174·53-s + 0.604·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 1.58T + 3T^{2} \) |
| 7 | \( 1 - 3.33T + 7T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 - 5.14T + 13T^{2} \) |
| 17 | \( 1 + 2.08T + 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 + 6.31T + 23T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 - 9.30T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 1.27T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.07T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 + 4.70T + 79T^{2} \) |
| 83 | \( 1 - 8.49T + 83T^{2} \) |
| 89 | \( 1 - 4.55T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.69556828341889551619682822232, −6.65017248368899116479020364722, −5.94785655288278307774009100743, −5.51060511255758605010054377389, −4.62078592839357454252972473165, −4.24071094923321419481615787640, −3.07452440650373950369574404391, −2.15319869568746851358864166920, −1.09497315953622543086315223578, 0,
1.09497315953622543086315223578, 2.15319869568746851358864166920, 3.07452440650373950369574404391, 4.24071094923321419481615787640, 4.62078592839357454252972473165, 5.51060511255758605010054377389, 5.94785655288278307774009100743, 6.65017248368899116479020364722, 7.69556828341889551619682822232
