| L(s) = 1 | + 5-s − 1.29·7-s − 11-s − 5.01·13-s + 3.29·17-s − 5.71·19-s + 5.71·23-s + 25-s + 2·29-s − 3.71·31-s − 1.29·35-s + 2·37-s + 2·41-s + 4.41·43-s − 9.71·47-s − 5.31·49-s + 4.59·53-s − 55-s + 3.71·59-s + 15.1·61-s − 5.01·65-s + 1.40·67-s − 16.0·71-s − 5.89·73-s + 1.29·77-s + 14.0·79-s − 7.52·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.491·7-s − 0.301·11-s − 1.38·13-s + 0.800·17-s − 1.31·19-s + 1.19·23-s + 0.200·25-s + 0.371·29-s − 0.666·31-s − 0.219·35-s + 0.328·37-s + 0.312·41-s + 0.672·43-s − 1.41·47-s − 0.758·49-s + 0.631·53-s − 0.134·55-s + 0.483·59-s + 1.93·61-s − 0.621·65-s + 0.171·67-s − 1.90·71-s − 0.690·73-s + 0.148·77-s + 1.57·79-s − 0.825·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593038896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593038896\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 1.29T + 7T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 + 5.71T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.71T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.41T + 43T^{2} \) |
| 47 | \( 1 + 9.71T + 47T^{2} \) |
| 53 | \( 1 - 4.59T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 1.40T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 6.82T + 97T^{2} \) |
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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
−7.75996345452145127973448516599, −7.11097472019912694059308531093, −6.52422286885806112015705653406, −5.73296700078319448249312901212, −5.06386354743250890686200088383, −4.42227795065607838240706659153, −3.37505538785569462798084646748, −2.66051125448216309949327349449, −1.91109944736295737812839854497, −0.60511841154966701003176262153,
0.60511841154966701003176262153, 1.91109944736295737812839854497, 2.66051125448216309949327349449, 3.37505538785569462798084646748, 4.42227795065607838240706659153, 5.06386354743250890686200088383, 5.73296700078319448249312901212, 6.52422286885806112015705653406, 7.11097472019912694059308531093, 7.75996345452145127973448516599
