| L(s) = 1 | − 5-s + 11-s − 8·17-s − 4·19-s − 4·23-s − 4·25-s − 5·29-s − 7·31-s − 8·37-s + 4·41-s − 10·43-s + 6·47-s − 53-s − 55-s − 9·59-s − 2·61-s − 2·67-s − 6·71-s − 2·73-s − 9·79-s + 3·83-s + 8·85-s − 6·89-s + 4·95-s + 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.94·17-s − 0.917·19-s − 0.834·23-s − 4/5·25-s − 0.928·29-s − 1.25·31-s − 1.31·37-s + 0.624·41-s − 1.52·43-s + 0.875·47-s − 0.137·53-s − 0.134·55-s − 1.17·59-s − 0.256·61-s − 0.244·67-s − 0.712·71-s − 0.234·73-s − 1.01·79-s + 0.329·83-s + 0.867·85-s − 0.635·89-s + 0.410·95-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-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
| \(L(1)\) |
\(\approx\) |
\(0.3023174818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3023174818\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.26796149669517, −14.80082526664060, −14.12632690750885, −13.59927790450871, −13.07943405612665, −12.60473739557571, −11.92387325049877, −11.48462348148807, −10.89911183039987, −10.52562332509967, −9.754925231182854, −9.077999955522964, −8.738241408353613, −8.135625725971625, −7.384148399372192, −6.983009949788786, −6.248587024322730, −5.792269478855358, −4.921205906880999, −4.235610139840471, −3.895064621155727, −3.086599076200652, −2.014463448289110, −1.787217205468160, −0.1986243258091365,
0.1986243258091365, 1.787217205468160, 2.014463448289110, 3.086599076200652, 3.895064621155727, 4.235610139840471, 4.921205906880999, 5.792269478855358, 6.248587024322730, 6.983009949788786, 7.384148399372192, 8.135625725971625, 8.738241408353613, 9.077999955522964, 9.754925231182854, 10.52562332509967, 10.89911183039987, 11.48462348148807, 11.92387325049877, 12.60473739557571, 13.07943405612665, 13.59927790450871, 14.12632690750885, 14.80082526664060, 15.26796149669517
