L(s) = 1 | + 3·3-s + 3·5-s + 7-s + 6·9-s + 3·11-s + 5·13-s + 9·15-s − 2·17-s + 4·19-s + 3·21-s + 4·23-s + 4·25-s + 9·27-s − 7·31-s + 9·33-s + 3·35-s + 2·37-s + 15·39-s + 8·41-s − 43-s + 18·45-s − 3·47-s + 49-s − 6·51-s + 9·53-s + 9·55-s + 12·57-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1.34·5-s + 0.377·7-s + 2·9-s + 0.904·11-s + 1.38·13-s + 2.32·15-s − 0.485·17-s + 0.917·19-s + 0.654·21-s + 0.834·23-s + 4/5·25-s + 1.73·27-s − 1.25·31-s + 1.56·33-s + 0.507·35-s + 0.328·37-s + 2.40·39-s + 1.24·41-s − 0.152·43-s + 2.68·45-s − 0.437·47-s + 1/7·49-s − 0.840·51-s + 1.23·53-s + 1.21·55-s + 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(9.612830760\) |
\(L(\frac12)\) |
\(\approx\) |
\(9.612830760\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.61408330207454, −14.01382962097981, −13.66137865132784, −13.18540874584272, −13.01130453257555, −12.09147898712303, −11.46255811674553, −10.73963707497550, −10.43077797374208, −9.495193791502353, −9.297790626305439, −8.960942669896910, −8.487655909299702, −7.726989722292273, −7.289186458955929, −6.607151830770315, −6.009602252833484, −5.489927252761990, −4.576439973738189, −4.038290443091339, −3.346868666423884, −2.876494922824304, −2.096355218772598, −1.524565562763103, −1.130986965877076,
1.130986965877076, 1.524565562763103, 2.096355218772598, 2.876494922824304, 3.346868666423884, 4.038290443091339, 4.576439973738189, 5.489927252761990, 6.009602252833484, 6.607151830770315, 7.289186458955929, 7.726989722292273, 8.487655909299702, 8.960942669896910, 9.297790626305439, 9.495193791502353, 10.43077797374208, 10.73963707497550, 11.46255811674553, 12.09147898712303, 13.01130453257555, 13.18540874584272, 13.66137865132784, 14.01382962097981, 14.61408330207454