| L(s) = 1 | + 0.831·2-s − 0.713·3-s − 1.30·4-s + 1.71·5-s − 0.593·6-s − 2.75·8-s − 2.49·9-s + 1.42·10-s + 6.30·11-s + 0.934·12-s − 2.82·13-s − 1.22·15-s + 0.331·16-s − 3.67·17-s − 2.07·18-s − 4.30·19-s − 2.24·20-s + 5.23·22-s − 1.07·23-s + 1.96·24-s − 2.05·25-s − 2.34·26-s + 3.91·27-s + 6.99·29-s − 1.01·30-s − 4.64·31-s + 5.77·32-s + ⋯ |
| L(s) = 1 | + 0.587·2-s − 0.412·3-s − 0.654·4-s + 0.767·5-s − 0.242·6-s − 0.972·8-s − 0.830·9-s + 0.450·10-s + 1.90·11-s + 0.269·12-s − 0.782·13-s − 0.316·15-s + 0.0828·16-s − 0.890·17-s − 0.488·18-s − 0.987·19-s − 0.502·20-s + 1.11·22-s − 0.223·23-s + 0.400·24-s − 0.411·25-s − 0.460·26-s + 0.754·27-s + 1.29·29-s − 0.185·30-s − 0.833·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677951067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677951067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.831T + 2T^{2} \) |
| 3 | \( 1 + 0.713T + 3T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 11 | \( 1 - 6.30T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 3.67T + 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 - 6.99T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 - 0.227T + 41T^{2} \) |
| 43 | \( 1 - 0.781T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 - 0.313T + 53T^{2} \) |
| 59 | \( 1 - 2.11T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 + 5.95T + 67T^{2} \) |
| 71 | \( 1 - 6.51T + 71T^{2} \) |
| 73 | \( 1 + 0.998T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 - 2.10T + 83T^{2} \) |
| 89 | \( 1 + 0.415T + 89T^{2} \) |
| 97 | \( 1 - 9.24T + 97T^{2} \) |
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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
−8.252157281226401191766672782132, −7.04091848763916386161046490829, −6.25599141144444294252091461227, −6.05261013928118932161864328343, −5.16831409590729634825302526085, −4.41927987351530910468381116672, −3.90286857027186513188543383034, −2.82081628789257217184154082048, −1.94701129347519371732436256544, −0.62330443792699848861641531259,
0.62330443792699848861641531259, 1.94701129347519371732436256544, 2.82081628789257217184154082048, 3.90286857027186513188543383034, 4.41927987351530910468381116672, 5.16831409590729634825302526085, 6.05261013928118932161864328343, 6.25599141144444294252091461227, 7.04091848763916386161046490829, 8.252157281226401191766672782132
