L(s) = 1 | − 3-s − 3·5-s + 9-s − 11-s + 3·13-s + 3·15-s − 19-s + 7·23-s + 4·25-s − 27-s − 6·29-s − 2·31-s + 33-s + 4·37-s − 3·39-s − 9·41-s + 43-s − 3·45-s − 10·47-s − 7·49-s − 2·53-s + 3·55-s + 57-s + 6·59-s + 12·61-s − 9·65-s + 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.774·15-s − 0.229·19-s + 1.45·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.174·33-s + 0.657·37-s − 0.480·39-s − 1.40·41-s + 0.152·43-s − 0.447·45-s − 1.45·47-s − 49-s − 0.274·53-s + 0.404·55-s + 0.132·57-s + 0.781·59-s + 1.53·61-s − 1.11·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.33520979276125, −16.01993468730280, −15.26880695654303, −15.00566149180374, −14.39947362650378, −13.33835697565993, −13.06855445723225, −12.52819960233137, −11.68269777759095, −11.38616646035349, −10.98483620757991, −10.34353289631360, −9.542180469758720, −8.890560650363459, −8.149425457605884, −7.829007475797260, −6.920841893350595, −6.645904910459483, −5.649246337786496, −5.059162759674450, −4.395099804755680, −3.616871266840290, −3.203644868826525, −1.951085675697849, −0.9215455229561430, 0,
0.9215455229561430, 1.951085675697849, 3.203644868826525, 3.616871266840290, 4.395099804755680, 5.059162759674450, 5.649246337786496, 6.645904910459483, 6.920841893350595, 7.829007475797260, 8.149425457605884, 8.890560650363459, 9.542180469758720, 10.34353289631360, 10.98483620757991, 11.38616646035349, 11.68269777759095, 12.52819960233137, 13.06855445723225, 13.33835697565993, 14.39947362650378, 15.00566149180374, 15.26880695654303, 16.01993468730280, 16.33520979276125