| L(s) = 1 | + 5-s + 5.12·7-s + 2·11-s − 5.12·13-s + 1.12·17-s − 5.12·19-s + 5.12·23-s + 25-s − 8.24·29-s − 7.12·31-s + 5.12·35-s + 5.12·37-s + 2·41-s + 6.24·43-s + 13.1·47-s + 19.2·49-s + 10·53-s + 2·55-s + 6·59-s + 2·61-s − 5.12·65-s + 6.24·67-s + 8·71-s − 4.24·73-s + 10.2·77-s − 4.87·79-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.93·7-s + 0.603·11-s − 1.42·13-s + 0.272·17-s − 1.17·19-s + 1.06·23-s + 0.200·25-s − 1.53·29-s − 1.27·31-s + 0.865·35-s + 0.842·37-s + 0.312·41-s + 0.952·43-s + 1.91·47-s + 2.74·49-s + 1.37·53-s + 0.269·55-s + 0.781·59-s + 0.256·61-s − 0.635·65-s + 0.763·67-s + 0.949·71-s − 0.496·73-s + 1.16·77-s − 0.548·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.786611436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.786611436\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.001644622231715371082345259487, −7.43286128492331394867783341088, −6.91970303859033905154740306463, −5.67011872211686513557467980031, −5.33631576215886632835076523247, −4.47703493935471993178056520398, −3.92245559356035957251808557618, −2.41364692880760398643922920785, −2.00563147968681177008788447608, −0.918468970184361290672860681967,
0.918468970184361290672860681967, 2.00563147968681177008788447608, 2.41364692880760398643922920785, 3.92245559356035957251808557618, 4.47703493935471993178056520398, 5.33631576215886632835076523247, 5.67011872211686513557467980031, 6.91970303859033905154740306463, 7.43286128492331394867783341088, 8.001644622231715371082345259487
