L(s) = 1 | − 5-s − 7-s − 11-s + 4·13-s − 6·17-s − 7·19-s − 23-s + 25-s + 6·29-s − 4·31-s + 35-s − 2·37-s − 9·41-s − 2·43-s + 7·47-s + 49-s − 5·53-s + 55-s + 7·59-s − 7·61-s − 4·65-s + 2·67-s + 4·71-s − 10·73-s + 77-s + 6·79-s + 16·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s + 1.10·13-s − 1.45·17-s − 1.60·19-s − 0.208·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.169·35-s − 0.328·37-s − 1.40·41-s − 0.304·43-s + 1.02·47-s + 1/7·49-s − 0.686·53-s + 0.134·55-s + 0.911·59-s − 0.896·61-s − 0.496·65-s + 0.244·67-s + 0.474·71-s − 1.17·73-s + 0.113·77-s + 0.675·79-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.64636197573836, −13.38139085389670, −12.95389383434358, −12.37537187013602, −11.99908163535025, −11.29653623659908, −10.91544083874523, −10.53071697204225, −10.11523030950007, −9.270514853004609, −8.867785618251875, −8.410183118544833, −8.133040370855080, −7.303463424916536, −6.716104963889051, −6.457720423103407, −5.917103782693349, −5.187795719684261, −4.529010756709068, −4.152235046383521, −3.546701593837654, −2.978142561452966, −2.182569090573861, −1.740926193790851, −0.6881394299182240, 0,
0.6881394299182240, 1.740926193790851, 2.182569090573861, 2.978142561452966, 3.546701593837654, 4.152235046383521, 4.529010756709068, 5.187795719684261, 5.917103782693349, 6.457720423103407, 6.716104963889051, 7.303463424916536, 8.133040370855080, 8.410183118544833, 8.867785618251875, 9.270514853004609, 10.11523030950007, 10.53071697204225, 10.91544083874523, 11.29653623659908, 11.99908163535025, 12.37537187013602, 12.95389383434358, 13.38139085389670, 13.64636197573836