| L(s) = 1 | − 2.11·3-s + 2.26·5-s + 4.06·7-s + 1.46·9-s + 1.00·11-s − 5.09·13-s − 4.78·15-s + 1.45·17-s − 6.53·19-s − 8.58·21-s + 4.54·23-s + 0.136·25-s + 3.25·27-s − 6.23·29-s + 10.1·31-s − 2.11·33-s + 9.21·35-s + 3.44·37-s + 10.7·39-s − 9.13·41-s − 2.78·43-s + 3.31·45-s − 6.12·47-s + 9.52·49-s − 3.07·51-s − 3.12·53-s + 2.27·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 1.01·5-s + 1.53·7-s + 0.486·9-s + 0.302·11-s − 1.41·13-s − 1.23·15-s + 0.352·17-s − 1.50·19-s − 1.87·21-s + 0.948·23-s + 0.0272·25-s + 0.625·27-s − 1.15·29-s + 1.81·31-s − 0.368·33-s + 1.55·35-s + 0.566·37-s + 1.72·39-s − 1.42·41-s − 0.424·43-s + 0.493·45-s − 0.892·47-s + 1.36·49-s − 0.430·51-s − 0.428·53-s + 0.306·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 2.11T + 3T^{2} \) |
| 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 - 1.00T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 19 | \( 1 + 6.53T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 + 2.78T + 43T^{2} \) |
| 47 | \( 1 + 6.12T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 0.606T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 6.43T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.44039906390074327001600447908, −6.52483692103614657570441647452, −6.12975627675524119014618592660, −5.26139433234333713970792776152, −4.87138390840754058186233181400, −4.38406021332297631995194387380, −2.89599594291914374386733693364, −1.97385220362694397831152403261, −1.34035791832109570084665018600, 0,
1.34035791832109570084665018600, 1.97385220362694397831152403261, 2.89599594291914374386733693364, 4.38406021332297631995194387380, 4.87138390840754058186233181400, 5.26139433234333713970792776152, 6.12975627675524119014618592660, 6.52483692103614657570441647452, 7.44039906390074327001600447908
