| L(s) = 1 | + 2.16·2-s + 3-s + 2.68·4-s + 2.16·6-s + 4.84·7-s + 1.48·8-s + 9-s + 3·11-s + 2.68·12-s − 0.519·13-s + 10.4·14-s − 2.16·16-s + 2.84·17-s + 2.16·18-s − 3.36·19-s + 4.84·21-s + 6.49·22-s − 6.80·23-s + 1.48·24-s − 1.12·26-s + 27-s + 13.0·28-s − 29-s − 7.64·32-s + 3·33-s + 6.16·34-s + 2.68·36-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.577·3-s + 1.34·4-s + 0.883·6-s + 1.83·7-s + 0.523·8-s + 0.333·9-s + 0.904·11-s + 0.774·12-s − 0.144·13-s + 2.80·14-s − 0.541·16-s + 0.690·17-s + 0.510·18-s − 0.772·19-s + 1.05·21-s + 1.38·22-s − 1.41·23-s + 0.302·24-s − 0.220·26-s + 0.192·27-s + 2.45·28-s − 0.185·29-s − 1.35·32-s + 0.522·33-s + 1.05·34-s + 0.447·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.156766204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.156766204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 7 | \( 1 - 4.84T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 0.519T + 13T^{2} \) |
| 17 | \( 1 - 2.84T + 17T^{2} \) |
| 19 | \( 1 + 3.36T + 19T^{2} \) |
| 23 | \( 1 + 6.80T + 23T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 7.84T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 3.44T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 5.36T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 8.58T + 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.813972663522099571633282197018, −8.246182014506372122085848519962, −7.43359338397312066616692158129, −6.56543446671410866068438699800, −5.63238044193880760565239356134, −4.92866357742522007968505500261, −4.16522623648030700895794611962, −3.61887807605568899539985127213, −2.30807327490443991618095563779, −1.60053996753598117862128092774,
1.60053996753598117862128092774, 2.30807327490443991618095563779, 3.61887807605568899539985127213, 4.16522623648030700895794611962, 4.92866357742522007968505500261, 5.63238044193880760565239356134, 6.56543446671410866068438699800, 7.43359338397312066616692158129, 8.246182014506372122085848519962, 8.813972663522099571633282197018
