| L(s) = 1 | − 2-s + 3-s + 4-s + 0.618·5-s − 6-s − 8-s + 9-s − 0.618·10-s − 4·11-s + 12-s + 1.61·13-s + 0.618·15-s + 16-s − 4.09·17-s − 18-s + 0.618·20-s + 4·22-s − 4·23-s − 24-s − 4.61·25-s − 1.61·26-s + 27-s − 8.85·29-s − 0.618·30-s + 1.52·31-s − 32-s − 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.276·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.195·10-s − 1.20·11-s + 0.288·12-s + 0.448·13-s + 0.159·15-s + 0.250·16-s − 0.992·17-s − 0.235·18-s + 0.138·20-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 0.923·25-s − 0.317·26-s + 0.192·27-s − 1.64·29-s − 0.112·30-s + 0.274·31-s − 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 4.09T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8.85T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 5.85T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 8.61T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 9.61T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
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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
−8.639104015306652654046022550134, −7.963052943678494101214506545683, −7.46426151540045305385790182310, −6.39515225940001507836900406469, −5.70406986855634074566283277660, −4.58739518288769493766585214665, −3.53448299680613733657951785136, −2.48337630399636066051273007451, −1.74217653237120005055592286485, 0,
1.74217653237120005055592286485, 2.48337630399636066051273007451, 3.53448299680613733657951785136, 4.58739518288769493766585214665, 5.70406986855634074566283277660, 6.39515225940001507836900406469, 7.46426151540045305385790182310, 7.963052943678494101214506545683, 8.639104015306652654046022550134
