| L(s) = 1 | − 2-s + 2.18·3-s + 4-s + 2.83·5-s − 2.18·6-s − 7-s − 8-s + 1.75·9-s − 2.83·10-s − 6.37·11-s + 2.18·12-s + 3.07·13-s + 14-s + 6.19·15-s + 16-s − 1.36·17-s − 1.75·18-s − 0.477·19-s + 2.83·20-s − 2.18·21-s + 6.37·22-s + 1.21·23-s − 2.18·24-s + 3.06·25-s − 3.07·26-s − 2.71·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.25·3-s + 0.5·4-s + 1.26·5-s − 0.890·6-s − 0.377·7-s − 0.353·8-s + 0.584·9-s − 0.897·10-s − 1.92·11-s + 0.629·12-s + 0.854·13-s + 0.267·14-s + 1.59·15-s + 0.250·16-s − 0.331·17-s − 0.413·18-s − 0.109·19-s + 0.634·20-s − 0.475·21-s + 1.35·22-s + 0.254·23-s − 0.445·24-s + 0.612·25-s − 0.603·26-s − 0.522·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 11 | \( 1 + 6.37T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 + 1.36T + 17T^{2} \) |
| 19 | \( 1 + 0.477T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 9.95T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 7.70T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 6.11T + 71T^{2} \) |
| 73 | \( 1 - 6.30T + 73T^{2} \) |
| 79 | \( 1 - 5.49T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 - 2.38T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−7.995109281267246814037652844850, −7.19462429234525552152043098103, −6.44106988292514171241802105988, −5.60411250444438655218616875060, −5.07285465431896800966537553385, −3.57288865214338256032721730471, −3.04280202861324771211273926107, −2.15703929798346562618816102073, −1.74139270446114097028201119033, 0,
1.74139270446114097028201119033, 2.15703929798346562618816102073, 3.04280202861324771211273926107, 3.57288865214338256032721730471, 5.07285465431896800966537553385, 5.60411250444438655218616875060, 6.44106988292514171241802105988, 7.19462429234525552152043098103, 7.995109281267246814037652844850
