| L(s) = 1 | + 3-s + 9-s − 3·11-s + 4·13-s + 4·19-s + 27-s + 9·29-s + 31-s − 3·33-s − 8·37-s + 4·39-s − 10·43-s − 6·47-s + 3·53-s + 4·57-s − 3·59-s − 10·61-s − 10·67-s + 6·71-s − 2·73-s + 79-s + 81-s − 9·83-s + 9·87-s + 6·89-s + 93-s + 97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.917·19-s + 0.192·27-s + 1.67·29-s + 0.179·31-s − 0.522·33-s − 1.31·37-s + 0.640·39-s − 1.52·43-s − 0.875·47-s + 0.412·53-s + 0.529·57-s − 0.390·59-s − 1.28·61-s − 1.22·67-s + 0.712·71-s − 0.234·73-s + 0.112·79-s + 1/9·81-s − 0.987·83-s + 0.964·87-s + 0.635·89-s + 0.103·93-s + 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.52924690942386, −13.88201541598823, −13.67826057544473, −13.24213043264336, −12.62793075222760, −12.03584649628341, −11.64453989077992, −10.90324734148740, −10.44783112469618, −10.03880319932558, −9.460220458941540, −8.740813307289221, −8.426017687851050, −7.947703206563187, −7.341745731580495, −6.741839883172119, −6.210653398633926, −5.524997610111617, −4.915340604192967, −4.446088019881702, −3.479536159829206, −3.218645500980598, −2.555243389877183, −1.671835412331636, −1.104983527966324, 0,
1.104983527966324, 1.671835412331636, 2.555243389877183, 3.218645500980598, 3.479536159829206, 4.446088019881702, 4.915340604192967, 5.524997610111617, 6.210653398633926, 6.741839883172119, 7.341745731580495, 7.947703206563187, 8.426017687851050, 8.740813307289221, 9.460220458941540, 10.03880319932558, 10.44783112469618, 10.90324734148740, 11.64453989077992, 12.03584649628341, 12.62793075222760, 13.24213043264336, 13.67826057544473, 13.88201541598823, 14.52924690942386
