| L(s) = 1 | + 1.56·2-s + 3-s + 0.438·4-s + 1.56·6-s − 1.56·7-s − 2.43·8-s + 9-s − 4.56·11-s + 0.438·12-s + 6.12·13-s − 2.43·14-s − 4.68·16-s + 2.12·17-s + 1.56·18-s − 6.56·19-s − 1.56·21-s − 7.12·22-s + 8.68·23-s − 2.43·24-s + 9.56·26-s + 27-s − 0.684·28-s − 9.80·31-s − 2.43·32-s − 4.56·33-s + 3.31·34-s + 0.438·36-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.219·4-s + 0.637·6-s − 0.590·7-s − 0.862·8-s + 0.333·9-s − 1.37·11-s + 0.126·12-s + 1.69·13-s − 0.651·14-s − 1.17·16-s + 0.514·17-s + 0.368·18-s − 1.50·19-s − 0.340·21-s − 1.51·22-s + 1.81·23-s − 0.497·24-s + 1.87·26-s + 0.192·27-s − 0.129·28-s − 1.76·31-s − 0.431·32-s − 0.794·33-s + 0.568·34-s + 0.0730·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 + 6.56T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 9.80T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 + 9.12T + 41T^{2} \) |
| 43 | \( 1 + 5.43T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 61 | \( 1 + 0.876T + 61T^{2} \) |
| 67 | \( 1 - 1.43T + 67T^{2} \) |
| 71 | \( 1 - 3.31T + 71T^{2} \) |
| 73 | \( 1 + 13T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 4.43T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
| 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
−8.187040246427812449290781318594, −6.98145439504953525605011390594, −6.50654698004632257742377498472, −5.52743854444902315141805122755, −5.09889151940817058206480370206, −4.05378997432840590860500573230, −3.37708994627210345304474404480, −2.91504553427368932357579158458, −1.71153751134847852864188010537, 0,
1.71153751134847852864188010537, 2.91504553427368932357579158458, 3.37708994627210345304474404480, 4.05378997432840590860500573230, 5.09889151940817058206480370206, 5.52743854444902315141805122755, 6.50654698004632257742377498472, 6.98145439504953525605011390594, 8.187040246427812449290781318594
