L(s) = 1 | + 5-s − 7-s − 4·13-s + 3·17-s − 4·19-s + 23-s + 25-s + 3·29-s − 7·31-s − 35-s + 11·37-s + 9·41-s − 4·43-s − 6·47-s − 6·49-s − 9·53-s − 3·59-s − 10·61-s − 4·65-s − 13·67-s − 9·71-s − 16·73-s + 8·79-s + 15·83-s + 3·85-s + 4·91-s − 4·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.10·13-s + 0.727·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.557·29-s − 1.25·31-s − 0.169·35-s + 1.80·37-s + 1.40·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s − 1.23·53-s − 0.390·59-s − 1.28·61-s − 0.496·65-s − 1.58·67-s − 1.06·71-s − 1.87·73-s + 0.900·79-s + 1.64·83-s + 0.325·85-s + 0.419·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.84969797978169092949434929016, −7.48164389444817512492949352085, −6.41212131854983538146997254868, −6.00527710158029841193586725649, −4.99284417818000870354653357291, −4.38950931844129619030038319213, −3.24096292811274637274852233408, −2.52652029975439196850705640063, −1.47207048091632143239880011558, 0,
1.47207048091632143239880011558, 2.52652029975439196850705640063, 3.24096292811274637274852233408, 4.38950931844129619030038319213, 4.99284417818000870354653357291, 6.00527710158029841193586725649, 6.41212131854983538146997254868, 7.48164389444817512492949352085, 7.84969797978169092949434929016