| L(s) = 1 | − 0.302·5-s + 7-s + 2.60·11-s + 0.605·13-s + 17-s − 6·19-s − 4.60·23-s − 4.90·25-s + 1.39·29-s − 10.3·31-s − 0.302·35-s − 7.21·37-s + 8.51·41-s − 0.697·43-s − 10·47-s + 49-s − 4.30·53-s − 0.788·55-s + 9.21·59-s − 0.697·61-s − 0.183·65-s + 6.30·67-s + 2·71-s + 4.51·73-s + 2.60·77-s − 6·79-s + 5.21·83-s + ⋯ |
| L(s) = 1 | − 0.135·5-s + 0.377·7-s + 0.785·11-s + 0.167·13-s + 0.242·17-s − 1.37·19-s − 0.960·23-s − 0.981·25-s + 0.258·29-s − 1.85·31-s − 0.0511·35-s − 1.18·37-s + 1.32·41-s − 0.106·43-s − 1.45·47-s + 0.142·49-s − 0.591·53-s − 0.106·55-s + 1.19·59-s − 0.0892·61-s − 0.0227·65-s + 0.770·67-s + 0.237·71-s + 0.528·73-s + 0.296·77-s − 0.675·79-s + 0.571·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 0.302T + 5T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 - 0.605T + 13T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 4.60T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 43 | \( 1 + 0.697T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 - 9.21T + 59T^{2} \) |
| 61 | \( 1 + 0.697T + 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.108979593817422247433184237943, −7.31729183768645289796526705167, −6.52869217625755531570987455456, −5.87862477031877733408370754336, −5.03944039429382285536203939970, −4.04402121556305279439084381138, −3.65415691747509280951990677050, −2.27249299282729789164636876950, −1.53104087031574070377530256741, 0,
1.53104087031574070377530256741, 2.27249299282729789164636876950, 3.65415691747509280951990677050, 4.04402121556305279439084381138, 5.03944039429382285536203939970, 5.87862477031877733408370754336, 6.52869217625755531570987455456, 7.31729183768645289796526705167, 8.108979593817422247433184237943
