| L(s) = 1 | − 2·3-s − 4·7-s + 9-s − 4·11-s − 4·13-s + 2·17-s + 19-s + 8·21-s + 4·27-s − 2·29-s + 8·31-s + 8·33-s + 4·37-s + 8·39-s + 6·41-s − 12·47-s + 9·49-s − 4·51-s − 8·53-s − 2·57-s − 2·61-s − 4·63-s − 14·67-s + 8·71-s − 6·73-s + 16·77-s − 4·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.485·17-s + 0.229·19-s + 1.74·21-s + 0.769·27-s − 0.371·29-s + 1.43·31-s + 1.39·33-s + 0.657·37-s + 1.28·39-s + 0.937·41-s − 1.75·47-s + 9/7·49-s − 0.560·51-s − 1.09·53-s − 0.264·57-s − 0.256·61-s − 0.503·63-s − 1.71·67-s + 0.949·71-s − 0.702·73-s + 1.82·77-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1188458472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1188458472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−15.19394348518764, −14.71591231526304, −13.98018756682862, −13.31559794970556, −12.91901870279024, −12.42431554497342, −12.02067488290074, −11.44267998801282, −10.78198404904434, −10.28851895601953, −9.792071601110367, −9.497592580862592, −8.608846935482575, −7.768184688858123, −7.466353470046430, −6.573798594771729, −6.254857656904135, −5.704810011249430, −4.989185055686299, −4.673604713965254, −3.616127713739708, −2.849738755409346, −2.541961021942302, −1.167690076783917, −0.1541146140036195,
0.1541146140036195, 1.167690076783917, 2.541961021942302, 2.849738755409346, 3.616127713739708, 4.673604713965254, 4.989185055686299, 5.704810011249430, 6.254857656904135, 6.573798594771729, 7.466353470046430, 7.768184688858123, 8.608846935482575, 9.497592580862592, 9.792071601110367, 10.28851895601953, 10.78198404904434, 11.44267998801282, 12.02067488290074, 12.42431554497342, 12.91901870279024, 13.31559794970556, 13.98018756682862, 14.71591231526304, 15.19394348518764
