L(s) = 1 | + 3-s + 7-s + 9-s + 6·13-s − 2·17-s + 8·19-s + 21-s + 8·23-s + 27-s − 2·29-s − 4·31-s + 2·37-s + 6·39-s − 6·41-s + 4·43-s + 8·47-s + 49-s − 2·51-s − 10·53-s + 8·57-s − 4·59-s − 2·61-s + 63-s + 4·67-s + 8·69-s + 12·71-s + 2·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.485·17-s + 1.83·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.328·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s + 1.05·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.963·69-s + 1.42·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.401452372\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.401452372\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.83791393615379549859057006204, −7.18351453958385978181168286303, −6.53856138228647749140615342574, −5.62163640176644849922538160577, −5.07121624971051172305654927599, −4.12213748834975248840560216817, −3.42741171252676750700509342988, −2.81487246730653660337724385092, −1.63750539552048323842418799980, −0.969993498707344654619235374340,
0.969993498707344654619235374340, 1.63750539552048323842418799980, 2.81487246730653660337724385092, 3.42741171252676750700509342988, 4.12213748834975248840560216817, 5.07121624971051172305654927599, 5.62163640176644849922538160577, 6.53856138228647749140615342574, 7.18351453958385978181168286303, 7.83791393615379549859057006204