| L(s) = 1 | − 3-s + 5-s − 2·9-s − 5·11-s − 7·13-s − 15-s + 3·17-s + 2·19-s + 8·23-s + 25-s + 5·27-s − 5·29-s + 10·31-s + 5·33-s + 4·37-s + 7·39-s + 6·41-s + 2·43-s − 2·45-s + 7·47-s − 3·51-s − 10·53-s − 5·55-s − 2·57-s + 10·59-s + 12·61-s − 7·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s − 1.94·13-s − 0.258·15-s + 0.727·17-s + 0.458·19-s + 1.66·23-s + 1/5·25-s + 0.962·27-s − 0.928·29-s + 1.79·31-s + 0.870·33-s + 0.657·37-s + 1.12·39-s + 0.937·41-s + 0.304·43-s − 0.298·45-s + 1.02·47-s − 0.420·51-s − 1.37·53-s − 0.674·55-s − 0.264·57-s + 1.30·59-s + 1.53·61-s − 0.868·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.087245619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087245619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−9.402195175064196345224163923961, −8.288225635235158209049417531779, −7.56793846342897653210506471906, −6.88382824698342573621557941164, −5.70455601651800109248396826390, −5.28603537534421022550155880259, −4.62651714719306486614775648654, −2.91065351671621099691740507495, −2.51581326623504893362388190719, −0.69036005295889719279144291289,
0.69036005295889719279144291289, 2.51581326623504893362388190719, 2.91065351671621099691740507495, 4.62651714719306486614775648654, 5.28603537534421022550155880259, 5.70455601651800109248396826390, 6.88382824698342573621557941164, 7.56793846342897653210506471906, 8.288225635235158209049417531779, 9.402195175064196345224163923961
