| L(s) = 1 | − 3-s + 3·7-s − 2·9-s − 6·11-s − 6·13-s + 2·19-s − 3·21-s − 23-s + 5·27-s + 2·29-s + 10·31-s + 6·33-s + 10·37-s + 6·39-s − 9·41-s − 4·43-s + 2·49-s − 2·57-s + 8·59-s + 6·61-s − 6·63-s − 5·67-s + 69-s − 6·71-s − 10·73-s − 18·77-s − 2·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s − 2/3·9-s − 1.80·11-s − 1.66·13-s + 0.458·19-s − 0.654·21-s − 0.208·23-s + 0.962·27-s + 0.371·29-s + 1.79·31-s + 1.04·33-s + 1.64·37-s + 0.960·39-s − 1.40·41-s − 0.609·43-s + 2/7·49-s − 0.264·57-s + 1.04·59-s + 0.768·61-s − 0.755·63-s − 0.610·67-s + 0.120·69-s − 0.712·71-s − 1.17·73-s − 2.05·77-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.37846794858100, −14.94797872614017, −14.38748746454502, −13.98100444179445, −13.22443669517255, −12.85223829798573, −11.99176890484293, −11.71061936155695, −11.35424620548775, −10.54184483891374, −10.09176758826114, −9.824293708637895, −8.652797288269467, −8.343907941688822, −7.690476577431212, −7.391754712407040, −6.503393169871080, −5.797477015736442, −5.185241288026594, −4.864025664575542, −4.433893384540936, −3.063840645524089, −2.658669399195726, −2.007788357671100, −0.8385136778949419, 0,
0.8385136778949419, 2.007788357671100, 2.658669399195726, 3.063840645524089, 4.433893384540936, 4.864025664575542, 5.185241288026594, 5.797477015736442, 6.503393169871080, 7.391754712407040, 7.690476577431212, 8.343907941688822, 8.652797288269467, 9.824293708637895, 10.09176758826114, 10.54184483891374, 11.35424620548775, 11.71061936155695, 11.99176890484293, 12.85223829798573, 13.22443669517255, 13.98100444179445, 14.38748746454502, 14.94797872614017, 15.37846794858100
