L(s) = 1 | + 3-s − 4·7-s + 9-s − 6·13-s + 2·17-s − 6·19-s − 4·21-s − 6·23-s + 27-s − 8·29-s + 8·31-s − 10·37-s − 6·39-s − 6·41-s + 4·43-s − 2·47-s + 9·49-s + 2·51-s + 6·53-s − 6·57-s + 12·59-s + 14·61-s − 4·63-s + 4·67-s − 6·69-s + 8·71-s + 4·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.66·13-s + 0.485·17-s − 1.37·19-s − 0.872·21-s − 1.25·23-s + 0.192·27-s − 1.48·29-s + 1.43·31-s − 1.64·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 0.794·57-s + 1.56·59-s + 1.79·61-s − 0.503·63-s + 0.488·67-s − 0.722·69-s + 0.949·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.047232650\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.047232650\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.71635886166269217487954107721, −6.83975352724545215218345826699, −6.61724757752237378082562017409, −5.64302623241328320441919433891, −4.94576320545378925752723026546, −3.89578101727303806105320012165, −3.55364370349829433417608891270, −2.46502462410993711014677827145, −2.11506528238477330572605931095, −0.43736330370584063404549956980,
0.43736330370584063404549956980, 2.11506528238477330572605931095, 2.46502462410993711014677827145, 3.55364370349829433417608891270, 3.89578101727303806105320012165, 4.94576320545378925752723026546, 5.64302623241328320441919433891, 6.61724757752237378082562017409, 6.83975352724545215218345826699, 7.71635886166269217487954107721