L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·13-s + 17-s − 4·19-s − 2·21-s + 6·23-s − 5·25-s − 27-s + 10·31-s + 8·37-s + 2·39-s − 6·41-s − 4·43-s − 12·47-s − 3·49-s − 51-s + 6·53-s + 4·57-s + 12·59-s − 8·61-s + 2·63-s + 4·67-s − 6·69-s − 6·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 0.436·21-s + 1.25·23-s − 25-s − 0.192·27-s + 1.79·31-s + 1.31·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s + 0.529·57-s + 1.56·59-s − 1.02·61-s + 0.251·63-s + 0.488·67-s − 0.722·69-s − 0.712·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.680060399\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.680060399\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.58513473767736, −13.27633639800600, −12.86391328838844, −12.10412831179730, −11.82353337764909, −11.32236116906784, −10.99206427051010, −10.21613279617364, −9.972301832638023, −9.450191198727720, −8.635355216494398, −8.278795528453021, −7.791634238818608, −7.196961287908376, −6.552443918403230, −6.286770268124805, −5.432279972513273, −5.079269295366628, −4.489563746598242, −4.119008650099517, −3.191253465535310, −2.621871950561876, −1.857723960688500, −1.264380725946096, −0.4419911156255434,
0.4419911156255434, 1.264380725946096, 1.857723960688500, 2.621871950561876, 3.191253465535310, 4.119008650099517, 4.489563746598242, 5.079269295366628, 5.432279972513273, 6.286770268124805, 6.552443918403230, 7.196961287908376, 7.791634238818608, 8.278795528453021, 8.635355216494398, 9.450191198727720, 9.972301832638023, 10.21613279617364, 10.99206427051010, 11.32236116906784, 11.82353337764909, 12.10412831179730, 12.86391328838844, 13.27633639800600, 13.58513473767736