| L(s) = 1 | + 2·5-s − 6·13-s − 2·17-s − 4·19-s + 4·23-s − 25-s + 10·29-s + 8·31-s + 6·37-s − 2·41-s − 4·43-s + 8·47-s + 10·53-s + 12·59-s + 2·61-s − 12·65-s + 12·67-s + 12·71-s + 14·73-s − 8·79-s + 12·83-s − 4·85-s − 2·89-s − 8·95-s − 10·97-s − 6·101-s − 2·109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1.37·53-s + 1.56·59-s + 0.256·61-s − 1.48·65-s + 1.46·67-s + 1.42·71-s + 1.63·73-s − 0.900·79-s + 1.31·83-s − 0.433·85-s − 0.211·89-s − 0.820·95-s − 1.01·97-s − 0.597·101-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.026379449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026379449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.555344573726047604841983027914, −7.933799074416126707455171537090, −6.77134895449812380720649851711, −6.59713630178964216066585844974, −5.44667611802556221133165304915, −4.86755050775725973175456781006, −4.05615503781601136533091987066, −2.58011476774497626276874510564, −2.34015073243752209924210835354, −0.829794148171823188778794120429,
0.829794148171823188778794120429, 2.34015073243752209924210835354, 2.58011476774497626276874510564, 4.05615503781601136533091987066, 4.86755050775725973175456781006, 5.44667611802556221133165304915, 6.59713630178964216066585844974, 6.77134895449812380720649851711, 7.933799074416126707455171537090, 8.555344573726047604841983027914
