| L(s) = 1 | − 2.52·3-s + 0.600·5-s + 7-s + 3.35·9-s + 5.46·11-s − 4.95·13-s − 1.51·15-s − 17-s − 4.17·19-s − 2.52·21-s + 2.69·23-s − 4.63·25-s − 0.903·27-s + 3.46·29-s + 2.09·31-s − 13.7·33-s + 0.600·35-s − 7.22·37-s + 12.5·39-s − 1.58·41-s − 4.85·43-s + 2.01·45-s + 0.590·47-s + 49-s + 2.52·51-s + 3.36·53-s + 3.28·55-s + ⋯ |
| L(s) = 1 | − 1.45·3-s + 0.268·5-s + 0.377·7-s + 1.11·9-s + 1.64·11-s − 1.37·13-s − 0.391·15-s − 0.242·17-s − 0.958·19-s − 0.550·21-s + 0.561·23-s − 0.927·25-s − 0.173·27-s + 0.642·29-s + 0.376·31-s − 2.39·33-s + 0.101·35-s − 1.18·37-s + 2.00·39-s − 0.248·41-s − 0.740·43-s + 0.300·45-s + 0.0861·47-s + 0.142·49-s + 0.353·51-s + 0.462·53-s + 0.442·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 - 0.600T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 + 7.22T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 - 0.590T + 47T^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 9.33T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−8.072213329952810570969300753953, −7.05825721939224024551788255501, −6.56244691349293551899407607052, −6.00877648506407968534269900763, −5.00668613439213439946976604765, −4.64583967108928208173105616143, −3.66405779871391285204216619620, −2.23942423771496700639826564725, −1.26873881143764741253856420567, 0,
1.26873881143764741253856420567, 2.23942423771496700639826564725, 3.66405779871391285204216619620, 4.64583967108928208173105616143, 5.00668613439213439946976604765, 6.00877648506407968534269900763, 6.56244691349293551899407607052, 7.05825721939224024551788255501, 8.072213329952810570969300753953
