| L(s) = 1 | + 3-s + 2·7-s + 9-s − 11-s + 2·13-s + 3·17-s + 2·19-s + 2·21-s − 23-s + 27-s + 6·29-s − 4·31-s − 33-s + 2·37-s + 2·39-s − 6·41-s − 10·43-s + 12·47-s − 3·49-s + 3·51-s − 6·53-s + 2·57-s − 12·59-s + 5·61-s + 2·63-s + 11·67-s − 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.727·17-s + 0.458·19-s + 0.436·21-s − 0.208·23-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 1.52·43-s + 1.75·47-s − 3/7·49-s + 0.420·51-s − 0.824·53-s + 0.264·57-s − 1.56·59-s + 0.640·61-s + 0.251·63-s + 1.34·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.706168702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.706168702\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.14619104717193, −13.53301485129693, −13.25184104414549, −12.54451429891696, −12.02840626553831, −11.64952559534770, −10.97666065391431, −10.56926148146299, −9.945404339103399, −9.616135928282311, −8.727923156947172, −8.571030444177663, −7.921059601519248, −7.543250984241203, −6.950606335849547, −6.294516839717437, −5.649065189675224, −5.138586921517614, −4.525301424463462, −3.987499213518900, −3.145563429799865, −2.933572860925162, −1.844303201284573, −1.516037498460205, −0.6132232132321234,
0.6132232132321234, 1.516037498460205, 1.844303201284573, 2.933572860925162, 3.145563429799865, 3.987499213518900, 4.525301424463462, 5.138586921517614, 5.649065189675224, 6.294516839717437, 6.950606335849547, 7.543250984241203, 7.921059601519248, 8.571030444177663, 8.727923156947172, 9.616135928282311, 9.945404339103399, 10.56926148146299, 10.97666065391431, 11.64952559534770, 12.02840626553831, 12.54451429891696, 13.25184104414549, 13.53301485129693, 14.14619104717193
