| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s + 6·13-s − 15-s − 2·17-s − 4·21-s + 8·23-s + 25-s + 27-s + 6·29-s + 4·35-s + 6·37-s + 6·39-s − 10·41-s + 4·43-s − 45-s − 8·47-s + 9·49-s − 2·51-s − 10·53-s + 6·61-s − 4·63-s − 6·65-s − 4·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 0.485·17-s − 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.676·35-s + 0.986·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 1.37·53-s + 0.768·61-s − 0.503·63-s − 0.744·65-s − 0.488·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.10763690873169, −13.42526529326296, −13.23687837927822, −12.86238547524223, −12.34247282876092, −11.61582043795430, −11.18416040486341, −10.66862105114492, −10.14199583712511, −9.582814193323233, −9.097025937410634, −8.641763104606202, −8.280397090421469, −7.571611908663986, −6.929396398178358, −6.440332267610607, −6.271827725813961, −5.359416147307799, −4.683748694612177, −4.052604582975076, −3.484098897895166, −3.058212072779881, −2.650435916849987, −1.530926953850505, −0.9351017457386448, 0,
0.9351017457386448, 1.530926953850505, 2.650435916849987, 3.058212072779881, 3.484098897895166, 4.052604582975076, 4.683748694612177, 5.359416147307799, 6.271827725813961, 6.440332267610607, 6.929396398178358, 7.571611908663986, 8.280397090421469, 8.641763104606202, 9.097025937410634, 9.582814193323233, 10.14199583712511, 10.66862105114492, 11.18416040486341, 11.61582043795430, 12.34247282876092, 12.86238547524223, 13.23687837927822, 13.42526529326296, 14.10763690873169
