| L(s) = 1 | + 5-s + 7-s + 2·13-s − 2·19-s + 25-s + 6·29-s + 4·31-s + 35-s − 4·37-s − 9·41-s + 43-s − 3·47-s − 6·49-s + 6·53-s − 61-s + 2·65-s + 13·67-s − 12·71-s − 16·73-s + 10·79-s + 12·83-s + 3·89-s + 2·91-s − 2·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.554·13-s − 0.458·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.169·35-s − 0.657·37-s − 1.40·41-s + 0.152·43-s − 0.437·47-s − 6/7·49-s + 0.824·53-s − 0.128·61-s + 0.248·65-s + 1.58·67-s − 1.42·71-s − 1.87·73-s + 1.12·79-s + 1.31·83-s + 0.317·89-s + 0.209·91-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−14.02287969623273, −13.68125683464184, −13.26232871018859, −12.71560103680186, −12.11144588578211, −11.71878173414437, −11.20647490269208, −10.56322116896899, −10.22359688961008, −9.796652350686883, −8.988481504105807, −8.668751075783251, −8.180843871610974, −7.645613243665281, −6.898299798879820, −6.439312406188483, −6.078519521717430, −5.191014944249878, −4.969849730697410, −4.224103895369679, −3.618626130039505, −2.947353539367366, −2.317545980888073, −1.598499486349677, −1.041080842390955, 0,
1.041080842390955, 1.598499486349677, 2.317545980888073, 2.947353539367366, 3.618626130039505, 4.224103895369679, 4.969849730697410, 5.191014944249878, 6.078519521717430, 6.439312406188483, 6.898299798879820, 7.645613243665281, 8.180843871610974, 8.668751075783251, 8.988481504105807, 9.796652350686883, 10.22359688961008, 10.56322116896899, 11.20647490269208, 11.71878173414437, 12.11144588578211, 12.71560103680186, 13.26232871018859, 13.68125683464184, 14.02287969623273
