| L(s) = 1 | + 2-s − 1.44·3-s + 4-s − 1.29·5-s − 1.44·6-s + 1.46·7-s + 8-s − 0.920·9-s − 1.29·10-s + 4.03·11-s − 1.44·12-s − 4.55·13-s + 1.46·14-s + 1.86·15-s + 16-s + 4.46·17-s − 0.920·18-s + 19-s − 1.29·20-s − 2.11·21-s + 4.03·22-s − 5.71·23-s − 1.44·24-s − 3.32·25-s − 4.55·26-s + 5.65·27-s + 1.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.832·3-s + 0.5·4-s − 0.579·5-s − 0.588·6-s + 0.554·7-s + 0.353·8-s − 0.306·9-s − 0.409·10-s + 1.21·11-s − 0.416·12-s − 1.26·13-s + 0.391·14-s + 0.482·15-s + 0.250·16-s + 1.08·17-s − 0.216·18-s + 0.229·19-s − 0.289·20-s − 0.461·21-s + 0.859·22-s − 1.19·23-s − 0.294·24-s − 0.664·25-s − 0.893·26-s + 1.08·27-s + 0.277·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 5 | \( 1 + 1.29T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 4.03T + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 29 | \( 1 + 2.18T + 29T^{2} \) |
| 31 | \( 1 - 2.09T + 31T^{2} \) |
| 37 | \( 1 - 0.932T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 5.26T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 5.85T + 67T^{2} \) |
| 71 | \( 1 + 9.87T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 6.53T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−7.34670051419170451539759772189, −6.70139096563388124732973077184, −5.94977794863308233811761928836, −5.36928954345595440365629042327, −4.74958430294467162227086738873, −4.00313341387436601686340669292, −3.34405380535419126476094014765, −2.26704506980568438905967288037, −1.26706232446887645713970544660, 0,
1.26706232446887645713970544660, 2.26704506980568438905967288037, 3.34405380535419126476094014765, 4.00313341387436601686340669292, 4.74958430294467162227086738873, 5.36928954345595440365629042327, 5.94977794863308233811761928836, 6.70139096563388124732973077184, 7.34670051419170451539759772189
