L(s) = 1 | − 1.60·2-s − 3-s + 0.567·4-s − 0.489·5-s + 1.60·6-s − 0.574·7-s + 2.29·8-s + 9-s + 0.784·10-s − 0.698·11-s − 0.567·12-s + 2.26·13-s + 0.919·14-s + 0.489·15-s − 4.81·16-s − 17-s − 1.60·18-s + 1.07·19-s − 0.277·20-s + 0.574·21-s + 1.11·22-s − 7.79·23-s − 2.29·24-s − 4.75·25-s − 3.62·26-s − 27-s − 0.325·28-s + ⋯ |
L(s) = 1 | − 1.13·2-s − 0.577·3-s + 0.283·4-s − 0.219·5-s + 0.654·6-s − 0.216·7-s + 0.811·8-s + 0.333·9-s + 0.248·10-s − 0.210·11-s − 0.163·12-s + 0.627·13-s + 0.245·14-s + 0.126·15-s − 1.20·16-s − 0.242·17-s − 0.377·18-s + 0.246·19-s − 0.0621·20-s + 0.125·21-s + 0.238·22-s − 1.62·23-s − 0.468·24-s − 0.951·25-s − 0.710·26-s − 0.192·27-s − 0.0615·28-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
\(L(1)\) |
\(\approx\) |
\(0.5481216112\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5481216112\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.60T + 2T^{2} \) |
| 5 | \( 1 + 0.489T + 5T^{2} \) |
| 7 | \( 1 + 0.574T + 7T^{2} \) |
| 11 | \( 1 + 0.698T + 11T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 + 0.352T + 37T^{2} \) |
| 41 | \( 1 - 9.93T + 41T^{2} \) |
| 43 | \( 1 - 7.78T + 43T^{2} \) |
| 47 | \( 1 - 2.72T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 3.05T + 67T^{2} \) |
| 71 | \( 1 - 3.33T + 71T^{2} \) |
| 73 | \( 1 + 6.93T + 73T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 6.31T + 89T^{2} \) |
| 97 | \( 1 + 1.51T + 97T^{2} \) |
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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
−8.164493226411327908023401328160, −8.089503276612199432050098868170, −7.13850817306353972363787378787, −6.29567041701065957474597383758, −5.71304732587954599426609384244, −4.50585627477543715397285775055, −4.08358261201473934330364966174, −2.72074176279853963425840896387, −1.56703531059202598892769410112, −0.54213775513460817595242206053,
0.54213775513460817595242206053, 1.56703531059202598892769410112, 2.72074176279853963425840896387, 4.08358261201473934330364966174, 4.50585627477543715397285775055, 5.71304732587954599426609384244, 6.29567041701065957474597383758, 7.13850817306353972363787378787, 8.089503276612199432050098868170, 8.164493226411327908023401328160