L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 4·11-s + 6·13-s − 4·15-s − 4·17-s − 6·19-s + 4·23-s − 25-s + 4·27-s − 6·29-s + 4·31-s + 8·33-s − 6·37-s − 12·39-s + 4·41-s − 12·43-s + 2·45-s + 12·47-s + 8·51-s + 6·53-s − 8·55-s + 12·57-s − 6·59-s − 6·61-s + 12·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 1.03·15-s − 0.970·17-s − 1.37·19-s + 0.834·23-s − 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 1.39·33-s − 0.986·37-s − 1.92·39-s + 0.624·41-s − 1.82·43-s + 0.298·45-s + 1.75·47-s + 1.12·51-s + 0.824·53-s − 1.07·55-s + 1.58·57-s − 0.781·59-s − 0.768·61-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.906690031836325849137256725986, −8.469211771245610089148154511743, −7.19082229586124578074034920219, −6.27553375344910138529733882784, −5.87291546812501719908913117409, −5.11410191404007082067605167401, −4.13955083379191747937773722173, −2.74638261558682217260390763801, −1.58470220771255174257119282588, 0,
1.58470220771255174257119282588, 2.74638261558682217260390763801, 4.13955083379191747937773722173, 5.11410191404007082067605167401, 5.87291546812501719908913117409, 6.27553375344910138529733882784, 7.19082229586124578074034920219, 8.469211771245610089148154511743, 8.906690031836325849137256725986