| L(s) = 1 | − 1.26·2-s − 0.395·4-s + 3.80·5-s + 2.18·7-s + 3.03·8-s − 4.81·10-s − 0.950·11-s − 0.168·13-s − 2.77·14-s − 3.05·16-s − 6.57·17-s − 1.15·19-s − 1.50·20-s + 1.20·22-s − 4.62·23-s + 9.44·25-s + 0.212·26-s − 0.866·28-s + 1.33·29-s − 2.20·32-s + 8.33·34-s + 8.31·35-s + 3.87·37-s + 1.46·38-s + 11.5·40-s − 0.328·41-s − 9.63·43-s + ⋯ |
| L(s) = 1 | − 0.895·2-s − 0.197·4-s + 1.69·5-s + 0.827·7-s + 1.07·8-s − 1.52·10-s − 0.286·11-s − 0.0466·13-s − 0.741·14-s − 0.762·16-s − 1.59·17-s − 0.264·19-s − 0.336·20-s + 0.256·22-s − 0.965·23-s + 1.88·25-s + 0.0417·26-s − 0.163·28-s + 0.248·29-s − 0.389·32-s + 1.42·34-s + 1.40·35-s + 0.636·37-s + 0.236·38-s + 1.82·40-s − 0.0512·41-s − 1.46·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 + 0.950T + 11T^{2} \) |
| 13 | \( 1 + 0.168T + 13T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 23 | \( 1 + 4.62T + 23T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 + 0.328T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 + 5.63T + 47T^{2} \) |
| 53 | \( 1 + 7.33T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 - 0.552T + 67T^{2} \) |
| 71 | \( 1 + 1.13T + 71T^{2} \) |
| 73 | \( 1 - 7.92T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 0.326T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.62828419865969585367706241032, −6.65368724860881623850338043410, −6.25211081321436286141597260314, −5.20959435017717382186753850686, −4.85801267067716722967129890339, −4.03384577661293230451039385354, −2.62136726943454580366477683341, −1.92227696121031647142033019546, −1.40187446541264422581988214842, 0,
1.40187446541264422581988214842, 1.92227696121031647142033019546, 2.62136726943454580366477683341, 4.03384577661293230451039385354, 4.85801267067716722967129890339, 5.20959435017717382186753850686, 6.25211081321436286141597260314, 6.65368724860881623850338043410, 7.62828419865969585367706241032
