| L(s) = 1 | − 1.24·2-s + 3-s − 0.455·4-s − 3.33·5-s − 1.24·6-s + 7-s + 3.05·8-s + 9-s + 4.14·10-s + 2.90·11-s − 0.455·12-s − 4.59·13-s − 1.24·14-s − 3.33·15-s − 2.88·16-s − 5.29·17-s − 1.24·18-s + 1.86·19-s + 1.51·20-s + 21-s − 3.61·22-s + 5.03·23-s + 3.05·24-s + 6.11·25-s + 5.70·26-s + 27-s − 0.455·28-s + ⋯ |
| L(s) = 1 | − 0.878·2-s + 0.577·3-s − 0.227·4-s − 1.49·5-s − 0.507·6-s + 0.377·7-s + 1.07·8-s + 0.333·9-s + 1.31·10-s + 0.876·11-s − 0.131·12-s − 1.27·13-s − 0.332·14-s − 0.860·15-s − 0.720·16-s − 1.28·17-s − 0.292·18-s + 0.428·19-s + 0.339·20-s + 0.218·21-s − 0.770·22-s + 1.04·23-s + 0.622·24-s + 1.22·25-s + 1.11·26-s + 0.192·27-s − 0.0860·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 + 4.59T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 - 5.03T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 - 0.968T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 0.666T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 9.88T + 71T^{2} \) |
| 73 | \( 1 + 2.57T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 0.493T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.580413127774544454982163327863, −7.71111113506960528873433446527, −7.40482016706651980532329718825, −6.63010885680894840288420615398, −4.92154005691844983084830983235, −4.48019072528173150371199271021, −3.70940389272013307579683991354, −2.57028550379554700689755255949, −1.24025408756021894088158879940, 0,
1.24025408756021894088158879940, 2.57028550379554700689755255949, 3.70940389272013307579683991354, 4.48019072528173150371199271021, 4.92154005691844983084830983235, 6.63010885680894840288420615398, 7.40482016706651980532329718825, 7.71111113506960528873433446527, 8.580413127774544454982163327863
