L(s) = 1 | + 2·3-s − 7-s + 9-s − 11-s − 13-s − 5·17-s − 2·21-s + 23-s − 5·25-s − 4·27-s − 3·29-s + 7·31-s − 2·33-s + 7·37-s − 2·39-s + 9·41-s − 6·43-s + 8·47-s + 49-s − 10·51-s − 6·53-s + 9·59-s + 7·61-s − 63-s − 9·67-s + 2·69-s − 10·71-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 1.21·17-s − 0.436·21-s + 0.208·23-s − 25-s − 0.769·27-s − 0.557·29-s + 1.25·31-s − 0.348·33-s + 1.15·37-s − 0.320·39-s + 1.40·41-s − 0.914·43-s + 1.16·47-s + 1/7·49-s − 1.40·51-s − 0.824·53-s + 1.17·59-s + 0.896·61-s − 0.125·63-s − 1.09·67-s + 0.240·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.414298313\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.414298313\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.90166518989972, −15.14311209688778, −14.92004313241838, −14.24041200505465, −13.66166868613489, −13.15063922677289, −12.96507104010715, −11.86707102516265, −11.60134847057725, −10.72820484700209, −10.21560649196211, −9.400315473122215, −9.199699360906955, −8.525454419779191, −7.804195927011135, −7.554986613578265, −6.602596882543644, −6.097712501500398, −5.287883695846615, −4.397431928015992, −3.919177166592873, −3.048763471257679, −2.489724358807069, −1.926483168750409, −0.5909980803709541,
0.5909980803709541, 1.926483168750409, 2.489724358807069, 3.048763471257679, 3.919177166592873, 4.397431928015992, 5.287883695846615, 6.097712501500398, 6.602596882543644, 7.554986613578265, 7.804195927011135, 8.525454419779191, 9.199699360906955, 9.400315473122215, 10.21560649196211, 10.72820484700209, 11.60134847057725, 11.86707102516265, 12.96507104010715, 13.15063922677289, 13.66166868613489, 14.24041200505465, 14.92004313241838, 15.14311209688778, 15.90166518989972