| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s − 2·13-s − 14-s + 16-s + 6·17-s + 18-s − 2·19-s − 21-s + 23-s + 24-s − 5·25-s − 2·26-s + 27-s − 28-s + 6·29-s + 8·31-s + 32-s + 6·34-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.218·21-s + 0.208·23-s + 0.204·24-s − 25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.79926087619322, −13.42766735414135, −12.95529109723233, −12.33111982893150, −12.09540318673562, −11.60701196654006, −10.99011483208472, −10.21295950279903, −10.00578871272323, −9.646511803062684, −8.850386662088248, −8.351985186400719, −7.765403729630524, −7.492603247148067, −6.806734117334906, −6.092928681287769, −6.014393411366662, −5.063185351027217, −4.623496079741052, −4.151114036377227, −3.388425328972183, −2.947156893787838, −2.566855129685421, −1.657883921991205, −1.102546799260359, 0,
1.102546799260359, 1.657883921991205, 2.566855129685421, 2.947156893787838, 3.388425328972183, 4.151114036377227, 4.623496079741052, 5.063185351027217, 6.014393411366662, 6.092928681287769, 6.806734117334906, 7.492603247148067, 7.765403729630524, 8.351985186400719, 8.850386662088248, 9.646511803062684, 10.00578871272323, 10.21295950279903, 10.99011483208472, 11.60701196654006, 12.09540318673562, 12.33111982893150, 12.95529109723233, 13.42766735414135, 13.79926087619322
