| L(s) = 1 | + 3·3-s − 4·7-s + 6·9-s − 5·11-s − 4·13-s + 6·17-s − 7·19-s − 12·21-s + 3·23-s + 9·27-s + 3·29-s + 4·31-s − 15·33-s − 8·37-s − 12·39-s − 6·41-s + 8·43-s − 6·47-s + 9·49-s + 18·51-s − 53-s − 21·57-s − 9·59-s + 2·61-s − 24·63-s − 10·67-s + 9·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.51·7-s + 2·9-s − 1.50·11-s − 1.10·13-s + 1.45·17-s − 1.60·19-s − 2.61·21-s + 0.625·23-s + 1.73·27-s + 0.557·29-s + 0.718·31-s − 2.61·33-s − 1.31·37-s − 1.92·39-s − 0.937·41-s + 1.21·43-s − 0.875·47-s + 9/7·49-s + 2.52·51-s − 0.137·53-s − 2.78·57-s − 1.17·59-s + 0.256·61-s − 3.02·63-s − 1.22·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496885354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496885354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| 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
−12.61129232546591, −12.28408693032318, −12.09322154909513, −10.88274032254263, −10.47187000286023, −10.24915498634936, −9.741717793154493, −9.386747526467290, −8.994197425415234, −8.334944028677557, −8.006217256006691, −7.672310097065633, −7.117919618465965, −6.610200415934811, −6.268306247309958, −5.343111429858532, −5.046119171504448, −4.384404523482381, −3.743772014546670, −3.258863714209889, −2.933303527538801, −2.469665364571363, −2.092722179422758, −1.213063744592740, −0.2714770834838810,
0.2714770834838810, 1.213063744592740, 2.092722179422758, 2.469665364571363, 2.933303527538801, 3.258863714209889, 3.743772014546670, 4.384404523482381, 5.046119171504448, 5.343111429858532, 6.268306247309958, 6.610200415934811, 7.117919618465965, 7.672310097065633, 8.006217256006691, 8.334944028677557, 8.994197425415234, 9.386747526467290, 9.741717793154493, 10.24915498634936, 10.47187000286023, 10.88274032254263, 12.09322154909513, 12.28408693032318, 12.61129232546591
