| L(s) = 1 | − 3-s − 5-s − 2·9-s + 3·11-s − 6·13-s + 15-s − 5·17-s + 19-s − 7·23-s − 4·25-s + 5·27-s + 2·29-s − 5·31-s − 3·33-s + 3·37-s + 6·39-s − 2·41-s − 4·43-s + 2·45-s + 5·47-s + 5·51-s − 53-s − 3·55-s − 57-s + 15·59-s − 5·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.904·11-s − 1.66·13-s + 0.258·15-s − 1.21·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s − 0.898·31-s − 0.522·33-s + 0.493·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 0.729·47-s + 0.700·51-s − 0.137·53-s − 0.404·55-s − 0.132·57-s + 1.95·59-s − 0.640·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−11.03349216671979865295647957473, −9.963841424895996623328721987432, −9.078576362092848814254068395687, −8.014488657450494746775996120076, −7.00425596608758194540819154944, −6.06267727883093747161626599029, −4.95144814234972906958385795845, −3.90401229468776288264995248683, −2.30393380848327924626818621905, 0,
2.30393380848327924626818621905, 3.90401229468776288264995248683, 4.95144814234972906958385795845, 6.06267727883093747161626599029, 7.00425596608758194540819154944, 8.014488657450494746775996120076, 9.078576362092848814254068395687, 9.963841424895996623328721987432, 11.03349216671979865295647957473
