L(s) = 1 | + 3-s + 4·7-s + 9-s − 6·13-s − 6·17-s − 2·19-s + 4·21-s + 4·23-s − 5·25-s + 27-s − 8·29-s + 6·31-s + 6·37-s − 6·39-s − 8·41-s + 6·43-s + 47-s + 9·49-s − 6·51-s − 2·53-s − 2·57-s − 12·59-s − 2·61-s + 4·63-s + 2·67-s + 4·69-s − 10·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.66·13-s − 1.45·17-s − 0.458·19-s + 0.872·21-s + 0.834·23-s − 25-s + 0.192·27-s − 1.48·29-s + 1.07·31-s + 0.986·37-s − 0.960·39-s − 1.24·41-s + 0.914·43-s + 0.145·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s − 0.264·57-s − 1.56·59-s − 0.256·61-s + 0.503·63-s + 0.244·67-s + 0.481·69-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 47 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.53980783083626262077277024423, −6.94529654303898782141955067816, −6.00918608288218482524230465434, −5.08153893199912648018478819298, −4.58249695784972388387234162590, −4.07631568296149717884837702441, −2.80463904775993784638062643067, −2.20551754130884966906924017014, −1.51839863908633248639104923158, 0,
1.51839863908633248639104923158, 2.20551754130884966906924017014, 2.80463904775993784638062643067, 4.07631568296149717884837702441, 4.58249695784972388387234162590, 5.08153893199912648018478819298, 6.00918608288218482524230465434, 6.94529654303898782141955067816, 7.53980783083626262077277024423