| L(s) = 1 | − 0.639·2-s + 3-s − 1.59·4-s − 1.53·5-s − 0.639·6-s − 2.74·7-s + 2.29·8-s + 9-s + 0.979·10-s + 4.38·11-s − 1.59·12-s + 2.19·13-s + 1.75·14-s − 1.53·15-s + 1.71·16-s + 17-s − 0.639·18-s + 4.55·19-s + 2.43·20-s − 2.74·21-s − 2.80·22-s − 7.35·23-s + 2.29·24-s − 2.65·25-s − 1.40·26-s + 27-s + 4.37·28-s + ⋯ |
| L(s) = 1 | − 0.452·2-s + 0.577·3-s − 0.795·4-s − 0.684·5-s − 0.260·6-s − 1.03·7-s + 0.811·8-s + 0.333·9-s + 0.309·10-s + 1.32·11-s − 0.459·12-s + 0.609·13-s + 0.469·14-s − 0.395·15-s + 0.428·16-s + 0.242·17-s − 0.150·18-s + 1.04·19-s + 0.544·20-s − 0.599·21-s − 0.597·22-s − 1.53·23-s + 0.468·24-s − 0.530·25-s − 0.275·26-s + 0.192·27-s + 0.826·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.116384158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116384158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.639T + 2T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 4.38T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + 7.35T + 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 - 3.78T + 41T^{2} \) |
| 43 | \( 1 - 3.27T + 43T^{2} \) |
| 47 | \( 1 + 5.56T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 2.49T + 61T^{2} \) |
| 67 | \( 1 + 0.208T + 67T^{2} \) |
| 71 | \( 1 - 5.71T + 71T^{2} \) |
| 73 | \( 1 + 1.55T + 73T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 - 8.42T + 89T^{2} \) |
| 97 | \( 1 - 1.90T + 97T^{2} \) |
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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
−8.502902120887404357378739681179, −7.83022562559898076391794642088, −7.24157413076469581884512293417, −6.30576234950497515558383277566, −5.56045121568908251922442163259, −4.27016767108030607769463051827, −3.81518138587009413884597881522, −3.27673310411858420999476051209, −1.77154837500844076236560625079, −0.65075938329851166663158904462,
0.65075938329851166663158904462, 1.77154837500844076236560625079, 3.27673310411858420999476051209, 3.81518138587009413884597881522, 4.27016767108030607769463051827, 5.56045121568908251922442163259, 6.30576234950497515558383277566, 7.24157413076469581884512293417, 7.83022562559898076391794642088, 8.502902120887404357378739681179
