| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 5·11-s − 13-s + 15-s + 17-s + 7·19-s + 2·21-s − 4·23-s + 25-s − 27-s − 4·29-s − 3·31-s − 5·33-s + 2·35-s + 11·37-s + 39-s + 4·41-s + 43-s − 45-s − 6·47-s − 3·49-s − 51-s + 10·53-s − 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s + 1.60·19-s + 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.538·31-s − 0.870·33-s + 0.338·35-s + 1.80·37-s + 0.160·39-s + 0.624·41-s + 0.152·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.140·51-s + 1.37·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.685447916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685447916\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.37837440150637, −14.10731011959051, −13.32785752472260, −12.80143758674907, −12.42352110306344, −11.71505607018298, −11.49885108891999, −11.16229309347853, −10.04188684418497, −9.944154654730268, −9.332886760198900, −8.897653445578837, −8.089382906079451, −7.354501039234874, −7.222054875185494, −6.375958805109364, −5.987439148426769, −5.462125846553426, −4.644978942631782, −4.070670367676184, −3.560353765254548, −2.990680696926463, −2.012354049282881, −1.166543728768108, −0.5348989898820951,
0.5348989898820951, 1.166543728768108, 2.012354049282881, 2.990680696926463, 3.560353765254548, 4.070670367676184, 4.644978942631782, 5.462125846553426, 5.987439148426769, 6.375958805109364, 7.222054875185494, 7.354501039234874, 8.089382906079451, 8.897653445578837, 9.332886760198900, 9.944154654730268, 10.04188684418497, 11.16229309347853, 11.49885108891999, 11.71505607018298, 12.42352110306344, 12.80143758674907, 13.32785752472260, 14.10731011959051, 14.37837440150637
