L(s) = 1 | + 5-s − 7-s − 4·13-s + 6·17-s − 2·19-s − 6·23-s + 25-s − 2·31-s − 35-s + 2·37-s + 6·41-s + 4·43-s + 49-s − 6·53-s − 12·59-s − 10·61-s − 4·65-s + 4·67-s − 12·71-s − 4·73-s − 8·79-s − 12·83-s + 6·85-s + 6·89-s + 4·91-s − 2·95-s + 8·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.10·13-s + 1.45·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s − 0.359·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s − 0.468·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.419·91-s − 0.205·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.62303324730615600555306548237, −7.42453209666565007953264828448, −6.18823131817684849597691730110, −5.89552237162032170404215264997, −4.94981086641525978002182301109, −4.18496018671806441822001764244, −3.19848947825628311453951559016, −2.43711426518665820924570102939, −1.42136912390124255279326542270, 0,
1.42136912390124255279326542270, 2.43711426518665820924570102939, 3.19848947825628311453951559016, 4.18496018671806441822001764244, 4.94981086641525978002182301109, 5.89552237162032170404215264997, 6.18823131817684849597691730110, 7.42453209666565007953264828448, 7.62303324730615600555306548237