L(s) = 1 | + 3-s − 4·5-s − 4·7-s + 9-s − 4·11-s + 6·13-s − 4·15-s − 4·19-s − 4·21-s + 8·23-s + 11·25-s + 27-s + 6·29-s − 4·33-s + 16·35-s − 6·37-s + 6·39-s + 6·43-s − 4·45-s − 8·47-s + 9·49-s + 12·53-s + 16·55-s − 4·57-s − 12·59-s + 2·61-s − 4·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.78·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 1.03·15-s − 0.917·19-s − 0.872·21-s + 1.66·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s − 0.696·33-s + 2.70·35-s − 0.986·37-s + 0.960·39-s + 0.914·43-s − 0.596·45-s − 1.16·47-s + 9/7·49-s + 1.64·53-s + 2.15·55-s − 0.529·57-s − 1.56·59-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.53323736401470341274720161375, −6.84381871792877211670172973029, −6.40140504696973363305283190301, −5.31110420013596944463042643574, −4.40837007179857655717407008559, −3.71659069609879246422725842967, −3.21061993224064181040619873254, −2.65529230514394038993222985574, −0.991680470465807326939902263567, 0,
0.991680470465807326939902263567, 2.65529230514394038993222985574, 3.21061993224064181040619873254, 3.71659069609879246422725842967, 4.40837007179857655717407008559, 5.31110420013596944463042643574, 6.40140504696973363305283190301, 6.84381871792877211670172973029, 7.53323736401470341274720161375