L(s) = 1 | + 4·4-s + 3·5-s + 7·7-s − 13·13-s + 16·16-s + 25·19-s + 12·20-s + 45·23-s − 16·25-s + 28·28-s + 33·29-s − 55·31-s + 21·35-s − 30·41-s − 5·43-s − 81·47-s + 49·49-s − 52·52-s − 15·53-s + 90·59-s + 64·64-s − 39·65-s + 29·73-s + 100·76-s + 67·79-s + 48·80-s + 159·83-s + ⋯ |
L(s) = 1 | + 4-s + 3/5·5-s + 7-s − 13-s + 16-s + 1.31·19-s + 3/5·20-s + 1.95·23-s − 0.639·25-s + 28-s + 1.13·29-s − 1.77·31-s + 3/5·35-s − 0.731·41-s − 0.116·43-s − 1.72·47-s + 49-s − 52-s − 0.283·53-s + 1.52·59-s + 64-s − 3/5·65-s + 0.397·73-s + 1.31·76-s + 0.848·79-s + 3/5·80-s + 1.91·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.073834177\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.073834177\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 - 3 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 25 T + p^{2} T^{2} \) |
| 23 | \( 1 - 45 T + p^{2} T^{2} \) |
| 29 | \( 1 - 33 T + p^{2} T^{2} \) |
| 31 | \( 1 + 55 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 + 30 T + p^{2} T^{2} \) |
| 43 | \( 1 + 5 T + p^{2} T^{2} \) |
| 47 | \( 1 + 81 T + p^{2} T^{2} \) |
| 53 | \( 1 + 15 T + p^{2} T^{2} \) |
| 59 | \( 1 - 90 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 29 T + p^{2} T^{2} \) |
| 79 | \( 1 - 67 T + p^{2} T^{2} \) |
| 83 | \( 1 - 159 T + p^{2} T^{2} \) |
| 89 | \( 1 + 165 T + p^{2} T^{2} \) |
| 97 | \( 1 + 131 T + p^{2} T^{2} \) |
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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
−10.07479443317493452079687038249, −9.327034255135507179103129718718, −8.189730772753198109597348663367, −7.35411260770961415297681626171, −6.74474863434218032927981903734, −5.45848331996311734276499980844, −4.98518240285083309561396704846, −3.32335232912160953566643698905, −2.27380262882479856976688217473, −1.27925895343918578529097919533,
1.27925895343918578529097919533, 2.27380262882479856976688217473, 3.32335232912160953566643698905, 4.98518240285083309561396704846, 5.45848331996311734276499980844, 6.74474863434218032927981903734, 7.35411260770961415297681626171, 8.189730772753198109597348663367, 9.327034255135507179103129718718, 10.07479443317493452079687038249