| L(s) = 1 | + 2-s + 3.18·3-s + 4-s − 2.45·5-s + 3.18·6-s + 1.12·7-s + 8-s + 7.14·9-s − 2.45·10-s + 5.61·11-s + 3.18·12-s − 1.19·13-s + 1.12·14-s − 7.82·15-s + 16-s + 0.269·17-s + 7.14·18-s − 4.36·19-s − 2.45·20-s + 3.59·21-s + 5.61·22-s − 23-s + 3.18·24-s + 1.03·25-s − 1.19·26-s + 13.1·27-s + 1.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.83·3-s + 0.5·4-s − 1.09·5-s + 1.30·6-s + 0.426·7-s + 0.353·8-s + 2.38·9-s − 0.777·10-s + 1.69·11-s + 0.919·12-s − 0.332·13-s + 0.301·14-s − 2.02·15-s + 0.250·16-s + 0.0653·17-s + 1.68·18-s − 1.00·19-s − 0.549·20-s + 0.785·21-s + 1.19·22-s − 0.208·23-s + 0.650·24-s + 0.207·25-s − 0.234·26-s + 2.53·27-s + 0.213·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.340213246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.340213246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 - 0.269T + 17T^{2} \) |
| 19 | \( 1 + 4.36T + 19T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 - 0.183T + 31T^{2} \) |
| 37 | \( 1 - 8.71T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 - 6.75T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 0.352T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 - 6.28T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.242124400063963973022991538169, −7.37597575279572179175516289131, −6.91105616558592049050767345218, −6.11130118086542968104026525554, −4.58181640649546890793198734733, −4.30817761125223598062820151938, −3.69437548338645865242348763423, −2.98480019183603848789492825866, −2.12348006857501457692140394727, −1.20588638053552248066069803563,
1.20588638053552248066069803563, 2.12348006857501457692140394727, 2.98480019183603848789492825866, 3.69437548338645865242348763423, 4.30817761125223598062820151938, 4.58181640649546890793198734733, 6.11130118086542968104026525554, 6.91105616558592049050767345218, 7.37597575279572179175516289131, 8.242124400063963973022991538169
