L(s) = 1 | + 2·5-s + 2·11-s + 4·13-s − 6·17-s − 8·19-s + 6·23-s − 25-s − 10·29-s + 4·31-s − 6·37-s + 6·41-s − 4·43-s − 8·47-s + 2·53-s + 4·55-s − 4·59-s + 8·61-s + 8·65-s + 8·67-s + 10·71-s + 4·73-s + 4·79-s + 12·83-s − 12·85-s + 14·89-s − 16·95-s + 4·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.603·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s + 1.25·23-s − 1/5·25-s − 1.85·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 0.274·53-s + 0.539·55-s − 0.520·59-s + 1.02·61-s + 0.992·65-s + 0.977·67-s + 1.18·71-s + 0.468·73-s + 0.450·79-s + 1.31·83-s − 1.30·85-s + 1.48·89-s − 1.64·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.38229613479058, −14.93722594986509, −14.50224965678178, −13.69044863408618, −13.34864430309462, −13.00558837795043, −12.45626981150889, −11.56053294738121, −11.03310369362934, −10.82338864917971, −10.09901162584372, −9.215863525288289, −9.167977427168900, −8.466277340256622, −7.930267734338607, −6.840598443733550, −6.584892389260741, −6.139052537214901, −5.371990211139407, −4.752671511429198, −3.945985073185899, −3.547670925787280, −2.352782883964225, −2.007183937727971, −1.179931545445440, 0,
1.179931545445440, 2.007183937727971, 2.352782883964225, 3.547670925787280, 3.945985073185899, 4.752671511429198, 5.371990211139407, 6.139052537214901, 6.584892389260741, 6.840598443733550, 7.930267734338607, 8.466277340256622, 9.167977427168900, 9.215863525288289, 10.09901162584372, 10.82338864917971, 11.03310369362934, 11.56053294738121, 12.45626981150889, 13.00558837795043, 13.34864430309462, 13.69044863408618, 14.50224965678178, 14.93722594986509, 15.38229613479058