L(s) = 1 | − 3-s − 3·7-s + 9-s − 2·11-s − 13-s − 2·17-s + 5·19-s + 3·21-s + 6·23-s − 27-s + 10·29-s + 3·31-s + 2·33-s − 2·37-s + 39-s − 8·41-s + 43-s + 2·47-s + 2·49-s + 2·51-s + 4·53-s − 5·57-s + 10·59-s + 7·61-s − 3·63-s − 3·67-s − 6·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.485·17-s + 1.14·19-s + 0.654·21-s + 1.25·23-s − 0.192·27-s + 1.85·29-s + 0.538·31-s + 0.348·33-s − 0.328·37-s + 0.160·39-s − 1.24·41-s + 0.152·43-s + 0.291·47-s + 2/7·49-s + 0.280·51-s + 0.549·53-s − 0.662·57-s + 1.30·59-s + 0.896·61-s − 0.377·63-s − 0.366·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.044139264\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.044139264\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.921309463296385728827644240175, −9.052471577485418050377330134019, −8.105315307318281005924759006469, −6.96660723932071234204269144809, −6.59754059703926529506773066348, −5.45352832460903594958062456545, −4.78985931636144963960080177818, −3.48045838663060024632885892599, −2.58914107657822994764341775063, −0.77482341387654784778243330666,
0.77482341387654784778243330666, 2.58914107657822994764341775063, 3.48045838663060024632885892599, 4.78985931636144963960080177818, 5.45352832460903594958062456545, 6.59754059703926529506773066348, 6.96660723932071234204269144809, 8.105315307318281005924759006469, 9.052471577485418050377330134019, 9.921309463296385728827644240175