L(s) = 1 | − 3-s − 4·7-s + 9-s − 4·11-s − 4·17-s + 4·21-s − 4·23-s − 27-s − 6·29-s + 4·31-s + 4·33-s + 8·37-s − 10·41-s − 4·43-s + 4·47-s + 9·49-s + 4·51-s + 12·53-s + 4·59-s + 2·61-s − 4·63-s + 4·67-s + 4·69-s + 8·73-s + 16·77-s − 12·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.970·17-s + 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s + 1.31·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s + 0.560·51-s + 1.64·53-s + 0.520·59-s + 0.256·61-s − 0.503·63-s + 0.488·67-s + 0.481·69-s + 0.936·73-s + 1.82·77-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−11.25119555178140085424640669637, −10.24713640301064257194420457710, −9.671728604717533470308800943454, −8.425273316407858927756969056783, −7.18372888449204428119332901427, −6.30782253827108715004959607025, −5.36371477889983409574050795450, −3.96453757467671714692485940103, −2.56579130253629562111590352122, 0,
2.56579130253629562111590352122, 3.96453757467671714692485940103, 5.36371477889983409574050795450, 6.30782253827108715004959607025, 7.18372888449204428119332901427, 8.425273316407858927756969056783, 9.671728604717533470308800943454, 10.24713640301064257194420457710, 11.25119555178140085424640669637