L(s) = 1 | − 3·5-s + 2·13-s − 2·17-s − 19-s − 4·23-s + 4·25-s − 8·29-s + 10·31-s + 37-s + 3·41-s + 2·43-s − 8·47-s − 5·53-s + 6·59-s − 6·65-s + 11·67-s + 71-s + 4·73-s − 8·79-s + 83-s + 6·85-s + 12·89-s + 3·95-s + 3·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.554·13-s − 0.485·17-s − 0.229·19-s − 0.834·23-s + 4/5·25-s − 1.48·29-s + 1.79·31-s + 0.164·37-s + 0.468·41-s + 0.304·43-s − 1.16·47-s − 0.686·53-s + 0.781·59-s − 0.744·65-s + 1.34·67-s + 0.118·71-s + 0.468·73-s − 0.900·79-s + 0.109·83-s + 0.650·85-s + 1.27·89-s + 0.307·95-s + 0.304·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.63871969324603, −13.17784665324627, −12.80443310189820, −12.10725541097390, −11.84406846947679, −11.28805060895612, −11.01117599752156, −10.46437593410227, −9.759520838699594, −9.420540582493653, −8.678288637669300, −8.225472008359764, −7.951344062617871, −7.422314263048258, −6.778700567235833, −6.351108218850137, −5.791468205082955, −5.053470885826470, −4.536256918521740, −3.880725420488744, −3.751850603299835, −2.926298122359058, −2.298233116321170, −1.526732671019888, −0.6920955676661823, 0,
0.6920955676661823, 1.526732671019888, 2.298233116321170, 2.926298122359058, 3.751850603299835, 3.880725420488744, 4.536256918521740, 5.053470885826470, 5.791468205082955, 6.351108218850137, 6.778700567235833, 7.422314263048258, 7.951344062617871, 8.225472008359764, 8.678288637669300, 9.420540582493653, 9.759520838699594, 10.46437593410227, 11.01117599752156, 11.28805060895612, 11.84406846947679, 12.10725541097390, 12.80443310189820, 13.17784665324627, 13.63871969324603