| L(s) = 1 | − 5-s + 2·7-s + 2·11-s + 2·13-s + 8·17-s + 4·19-s − 23-s + 25-s − 2·29-s + 8·31-s − 2·35-s + 8·37-s − 2·41-s + 8·43-s − 8·47-s − 3·49-s + 6·53-s − 2·55-s + 8·59-s − 8·61-s − 2·65-s − 16·67-s − 8·71-s − 10·73-s + 4·77-s + 10·79-s − 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s + 1.94·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.338·35-s + 1.31·37-s − 0.312·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 0.824·53-s − 0.269·55-s + 1.04·59-s − 1.02·61-s − 0.248·65-s − 1.95·67-s − 0.949·71-s − 1.17·73-s + 0.455·77-s + 1.12·79-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.690677968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.690677968\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−7.61505212632789996113508767499, −7.49773612204798321584851241080, −6.28988115965241922050223454588, −5.80183476107929501091250268076, −4.95972647220750506084694373024, −4.28112548281888838223282221110, −3.48929419747191510567134442335, −2.82575453415142609975188364578, −1.49783985006865674797499573616, −0.917192495086516521312324524460,
0.917192495086516521312324524460, 1.49783985006865674797499573616, 2.82575453415142609975188364578, 3.48929419747191510567134442335, 4.28112548281888838223282221110, 4.95972647220750506084694373024, 5.80183476107929501091250268076, 6.28988115965241922050223454588, 7.49773612204798321584851241080, 7.61505212632789996113508767499
