| L(s) = 1 | + 2-s − 2.02·3-s + 4-s + 3.58·5-s − 2.02·6-s + 0.649·7-s + 8-s + 1.08·9-s + 3.58·10-s + 0.375·11-s − 2.02·12-s + 0.649·14-s − 7.24·15-s + 16-s − 3.23·17-s + 1.08·18-s + 19-s + 3.58·20-s − 1.31·21-s + 0.375·22-s + 2.27·23-s − 2.02·24-s + 7.82·25-s + 3.86·27-s + 0.649·28-s − 4.42·29-s − 7.24·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.16·3-s + 0.5·4-s + 1.60·5-s − 0.825·6-s + 0.245·7-s + 0.353·8-s + 0.362·9-s + 1.13·10-s + 0.113·11-s − 0.583·12-s + 0.173·14-s − 1.86·15-s + 0.250·16-s − 0.784·17-s + 0.256·18-s + 0.229·19-s + 0.800·20-s − 0.286·21-s + 0.0800·22-s + 0.473·23-s − 0.412·24-s + 1.56·25-s + 0.743·27-s + 0.122·28-s − 0.822·29-s − 1.32·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997189441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997189441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 - 3.58T + 5T^{2} \) |
| 7 | \( 1 - 0.649T + 7T^{2} \) |
| 11 | \( 1 - 0.375T + 11T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 23 | \( 1 - 2.27T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 + 4.13T + 31T^{2} \) |
| 37 | \( 1 - 0.0221T + 37T^{2} \) |
| 41 | \( 1 - 8.86T + 41T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 - 7.39T + 59T^{2} \) |
| 61 | \( 1 - 1.47T + 61T^{2} \) |
| 67 | \( 1 + 3.58T + 67T^{2} \) |
| 71 | \( 1 - 8.57T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 4.59T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 4.35T + 97T^{2} \) |
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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
−7.80224159020851771996817204079, −6.86232019418750741483052026358, −6.44002945137065253292591483778, −5.71182682017964053240562498672, −5.36903108744682927192006255738, −4.74167249809699108306538315198, −3.77684529843724181509678697842, −2.58579140683288560369410093462, −1.93747827101251020907850302872, −0.871123493978878297107617540155,
0.871123493978878297107617540155, 1.93747827101251020907850302872, 2.58579140683288560369410093462, 3.77684529843724181509678697842, 4.74167249809699108306538315198, 5.36903108744682927192006255738, 5.71182682017964053240562498672, 6.44002945137065253292591483778, 6.86232019418750741483052026358, 7.80224159020851771996817204079
