| L(s) = 1 | − 2.28·2-s + 3-s + 3.21·4-s − 2.28·6-s + 7-s − 2.77·8-s + 9-s + 0.505·11-s + 3.21·12-s + 13-s − 2.28·14-s − 0.0864·16-s − 2.34·17-s − 2.28·18-s − 7.65·19-s + 21-s − 1.15·22-s − 3.87·23-s − 2.77·24-s − 2.28·26-s + 27-s + 3.21·28-s + 4.51·29-s − 5.77·31-s + 5.75·32-s + 0.505·33-s + 5.34·34-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 0.577·3-s + 1.60·4-s − 0.932·6-s + 0.377·7-s − 0.982·8-s + 0.333·9-s + 0.152·11-s + 0.928·12-s + 0.277·13-s − 0.610·14-s − 0.0216·16-s − 0.567·17-s − 0.538·18-s − 1.75·19-s + 0.218·21-s − 0.245·22-s − 0.808·23-s − 0.567·24-s − 0.447·26-s + 0.192·27-s + 0.607·28-s + 0.839·29-s − 1.03·31-s + 1.01·32-s + 0.0879·33-s + 0.916·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 11 | \( 1 - 0.505T + 11T^{2} \) |
| 17 | \( 1 + 2.34T + 17T^{2} \) |
| 19 | \( 1 + 7.65T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 - 4.51T + 29T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 - 0.538T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 1.39T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 2.95T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
| 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
−7.80038483328583148208221400225, −7.26648190186727395750165537285, −6.48036925996083434532725374179, −5.85829622473331384711872686083, −4.49236304977322013517821075249, −4.03126957804955666134647461349, −2.66916502315050766973428129285, −2.09520223521070325632832171423, −1.23444191774232137957432120299, 0,
1.23444191774232137957432120299, 2.09520223521070325632832171423, 2.66916502315050766973428129285, 4.03126957804955666134647461349, 4.49236304977322013517821075249, 5.85829622473331384711872686083, 6.48036925996083434532725374179, 7.26648190186727395750165537285, 7.80038483328583148208221400225
