| L(s) = 1 | − 5.13·5-s − 0.864·7-s − 11·11-s + 51.6·13-s + 52.8·17-s − 56.2·19-s − 211.·23-s − 98.6·25-s − 174.·29-s + 211.·31-s + 4.43·35-s − 86.9·37-s − 151.·41-s − 74·43-s − 163.·47-s − 342.·49-s − 539.·53-s + 56.4·55-s − 365.·59-s + 499.·61-s − 265.·65-s + 189.·67-s − 751.·71-s + 352.·73-s + 9.50·77-s − 319.·79-s − 811.·83-s + ⋯ |
| L(s) = 1 | − 0.459·5-s − 0.0466·7-s − 0.301·11-s + 1.10·13-s + 0.753·17-s − 0.679·19-s − 1.92·23-s − 0.789·25-s − 1.11·29-s + 1.22·31-s + 0.0214·35-s − 0.386·37-s − 0.575·41-s − 0.262·43-s − 0.507·47-s − 0.997·49-s − 1.39·53-s + 0.138·55-s − 0.805·59-s + 1.04·61-s − 0.506·65-s + 0.345·67-s − 1.25·71-s + 0.564·73-s + 0.0140·77-s − 0.455·79-s − 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 5.13T + 125T^{2} \) |
| 7 | \( 1 + 0.864T + 343T^{2} \) |
| 13 | \( 1 - 51.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 56.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 211.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 151.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 74T + 7.95e4T^{2} \) |
| 47 | \( 1 + 163.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 539.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 365.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 189.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 751.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 352.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 319.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 811.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 551.T + 9.12e5T^{2} \) |
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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
−10.40529560901769179360940352977, −9.602780152248101710406336583547, −8.305205710953796645705507284429, −7.87811740800547413327798666806, −6.50910591695764302787481839212, −5.66994616577643587810566789130, −4.28351314502518495303545676599, −3.36512440568639058159585179703, −1.74515037331068871195572042549, 0,
1.74515037331068871195572042549, 3.36512440568639058159585179703, 4.28351314502518495303545676599, 5.66994616577643587810566789130, 6.50910591695764302787481839212, 7.87811740800547413327798666806, 8.305205710953796645705507284429, 9.602780152248101710406336583547, 10.40529560901769179360940352977
