L(s) = 1 | + 3-s + 7-s + 9-s + 11-s + 2·13-s + 6·17-s + 4·19-s + 21-s + 27-s + 6·29-s + 33-s + 10·37-s + 2·39-s + 2·41-s + 4·43-s + 49-s + 6·51-s − 6·53-s + 4·57-s + 4·59-s − 2·61-s + 63-s − 4·67-s + 16·71-s − 2·73-s + 77-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.192·27-s + 1.11·29-s + 0.174·33-s + 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.125·63-s − 0.488·67-s + 1.89·71-s − 0.234·73-s + 0.113·77-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.202271330\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.202271330\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.80602779695841, −13.59637909248009, −12.83129888751489, −12.27750905416438, −12.06723678010744, −11.23082556166929, −11.03892288498144, −10.25880206055104, −9.772815211527703, −9.430943616504904, −8.847881184575841, −8.182198968895868, −7.881380052704605, −7.463297611220402, −6.713288599444173, −6.223852924629062, −5.560295654906467, −5.114732670773128, −4.338195394487217, −3.916617212793397, −3.153280168669769, −2.827500121768353, −1.971523535769558, −1.163625489974577, −0.8263610132535359,
0.8263610132535359, 1.163625489974577, 1.971523535769558, 2.827500121768353, 3.153280168669769, 3.916617212793397, 4.338195394487217, 5.114732670773128, 5.560295654906467, 6.223852924629062, 6.713288599444173, 7.463297611220402, 7.881380052704605, 8.182198968895868, 8.847881184575841, 9.430943616504904, 9.772815211527703, 10.25880206055104, 11.03892288498144, 11.23082556166929, 12.06723678010744, 12.27750905416438, 12.83129888751489, 13.59637909248009, 13.80602779695841