| L(s) = 1 | − 2-s − 1.88·3-s + 4-s + 5-s + 1.88·6-s − 3.45·7-s − 8-s + 0.571·9-s − 10-s + 11-s − 1.88·12-s − 1.07·13-s + 3.45·14-s − 1.88·15-s + 16-s + 4.59·17-s − 0.571·18-s − 6.77·19-s + 20-s + 6.53·21-s − 22-s + 8.70·23-s + 1.88·24-s + 25-s + 1.07·26-s + 4.58·27-s − 3.45·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.09·3-s + 0.5·4-s + 0.447·5-s + 0.771·6-s − 1.30·7-s − 0.353·8-s + 0.190·9-s − 0.316·10-s + 0.301·11-s − 0.545·12-s − 0.298·13-s + 0.923·14-s − 0.487·15-s + 0.250·16-s + 1.11·17-s − 0.134·18-s − 1.55·19-s + 0.223·20-s + 1.42·21-s − 0.213·22-s + 1.81·23-s + 0.385·24-s + 0.200·25-s + 0.211·26-s + 0.883·27-s − 0.653·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4745444669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4745444669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.88T + 3T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 6.77T + 19T^{2} \) |
| 23 | \( 1 - 8.70T + 23T^{2} \) |
| 29 | \( 1 + 5.36T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 7.42T + 41T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 7.57T + 61T^{2} \) |
| 67 | \( 1 - 6.54T + 67T^{2} \) |
| 71 | \( 1 + 9.42T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 2.04T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.529330270673319801089433283185, −7.22305518659260408539536448137, −6.92502976644476902386161979802, −6.09038530600493259050210667168, −5.68002712638149775615910596381, −4.83055347940226097487061392558, −3.58296639078610139929703141964, −2.84337334441430772334286771202, −1.64113162598623176045817451033, −0.44519608781075129718803084684,
0.44519608781075129718803084684, 1.64113162598623176045817451033, 2.84337334441430772334286771202, 3.58296639078610139929703141964, 4.83055347940226097487061392558, 5.68002712638149775615910596381, 6.09038530600493259050210667168, 6.92502976644476902386161979802, 7.22305518659260408539536448137, 8.529330270673319801089433283185
