| L(s) = 1 | + 3.39·2-s − 3·3-s + 3.52·4-s − 6.09·5-s − 10.1·6-s + 1.40·7-s − 15.1·8-s + 9·9-s − 20.6·10-s + 1.38·11-s − 10.5·12-s − 63.1·13-s + 4.76·14-s + 18.2·15-s − 79.7·16-s − 121.·17-s + 30.5·18-s − 44.5·19-s − 21.4·20-s − 4.20·21-s + 4.69·22-s + 138.·23-s + 45.5·24-s − 87.8·25-s − 214.·26-s − 27·27-s + 4.94·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 0.577·3-s + 0.440·4-s − 0.545·5-s − 0.692·6-s + 0.0757·7-s − 0.671·8-s + 0.333·9-s − 0.654·10-s + 0.0378·11-s − 0.254·12-s − 1.34·13-s + 0.0909·14-s + 0.314·15-s − 1.24·16-s − 1.72·17-s + 0.400·18-s − 0.537·19-s − 0.240·20-s − 0.0437·21-s + 0.0454·22-s + 1.25·23-s + 0.387·24-s − 0.702·25-s − 1.61·26-s − 0.192·27-s + 0.0333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 3.39T + 8T^{2} \) |
| 5 | \( 1 + 6.09T + 125T^{2} \) |
| 7 | \( 1 - 1.40T + 343T^{2} \) |
| 11 | \( 1 - 1.38T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6.31T + 2.43e4T^{2} \) |
| 31 | \( 1 - 246.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 110.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 156.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 484.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 606.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 724.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 145.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 858.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 6.88T + 7.04e5T^{2} \) |
| 97 | \( 1 + 540.T + 9.12e5T^{2} \) |
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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
−11.79714379223801896226705076279, −11.28139985205193439746675202113, −9.879210385815398916744400220468, −8.644803837501374254809220514966, −7.16773300611028429954840923658, −6.21312739133989569841835040733, −4.86399705481368748169217193795, −4.29183320713496553631172998655, −2.64189293885785430780595029057, 0,
2.64189293885785430780595029057, 4.29183320713496553631172998655, 4.86399705481368748169217193795, 6.21312739133989569841835040733, 7.16773300611028429954840923658, 8.644803837501374254809220514966, 9.879210385815398916744400220468, 11.28139985205193439746675202113, 11.79714379223801896226705076279
