L(s) = 1 | − 5-s − 4·11-s + 2·17-s + 6·19-s − 6·23-s + 25-s − 2·31-s + 2·37-s + 2·41-s + 4·43-s + 8·47-s − 10·53-s + 4·55-s + 4·59-s + 2·61-s + 12·67-s − 8·71-s − 8·73-s − 8·79-s − 4·83-s − 2·85-s + 10·89-s − 6·95-s + 4·97-s − 18·101-s − 4·103-s − 6·107-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.485·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 0.359·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s − 1.37·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 1.46·67-s − 0.949·71-s − 0.936·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s + 1.05·89-s − 0.615·95-s + 0.406·97-s − 1.79·101-s − 0.394·103-s − 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.62630071177267224012857264302, −6.87223153992264215008097418718, −5.81788120339171454600606963058, −5.46992471391578238282285336987, −4.60969839506760961675023504870, −3.83614389125609158470044894104, −3.05287621398091923524388935628, −2.32699827494530495543179701615, −1.15012981376703002607834362817, 0,
1.15012981376703002607834362817, 2.32699827494530495543179701615, 3.05287621398091923524388935628, 3.83614389125609158470044894104, 4.60969839506760961675023504870, 5.46992471391578238282285336987, 5.81788120339171454600606963058, 6.87223153992264215008097418718, 7.62630071177267224012857264302