| L(s) = 1 | + 2-s + 4-s − 0.683·5-s − 0.639·7-s + 8-s − 0.683·10-s − 1.37·11-s + 1.22·13-s − 0.639·14-s + 16-s − 2.93·17-s − 1.25·19-s − 0.683·20-s − 1.37·22-s + 7.55·23-s − 4.53·25-s + 1.22·26-s − 0.639·28-s − 2.51·29-s + 5.78·31-s + 32-s − 2.93·34-s + 0.436·35-s − 2.36·37-s − 1.25·38-s − 0.683·40-s − 1.79·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.305·5-s − 0.241·7-s + 0.353·8-s − 0.216·10-s − 0.415·11-s + 0.340·13-s − 0.170·14-s + 0.250·16-s − 0.712·17-s − 0.288·19-s − 0.152·20-s − 0.293·22-s + 1.57·23-s − 0.906·25-s + 0.240·26-s − 0.120·28-s − 0.466·29-s + 1.03·31-s + 0.176·32-s − 0.503·34-s + 0.0738·35-s − 0.389·37-s − 0.203·38-s − 0.108·40-s − 0.279·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 0.683T + 5T^{2} \) |
| 7 | \( 1 + 0.639T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 2.51T + 29T^{2} \) |
| 31 | \( 1 - 5.78T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 3.25T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 1.02T + 61T^{2} \) |
| 67 | \( 1 - 8.00T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 2.29T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 - 0.495T + 97T^{2} \) |
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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
−7.28418803949060901657172348088, −6.72737885958308634395614094201, −6.14972464759606050283047754122, −5.21298207053719401269488587850, −4.75844618633161486138974432185, −3.85944455099924691703773738802, −3.22689379766815340820490517518, −2.41960644768587308284598281818, −1.41582879122772396150512583547, 0,
1.41582879122772396150512583547, 2.41960644768587308284598281818, 3.22689379766815340820490517518, 3.85944455099924691703773738802, 4.75844618633161486138974432185, 5.21298207053719401269488587850, 6.14972464759606050283047754122, 6.72737885958308634395614094201, 7.28418803949060901657172348088
