L(s) = 1 | − 4·2-s + 8·4-s − 4·5-s + 16·10-s − 62·11-s + 62·13-s − 64·16-s + 84·17-s − 100·19-s − 32·20-s + 248·22-s + 42·23-s − 109·25-s − 248·26-s + 10·29-s + 48·31-s + 256·32-s − 336·34-s − 246·37-s + 400·38-s − 248·41-s + 68·43-s − 496·44-s − 168·46-s + 324·47-s + 436·50-s + 496·52-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.357·5-s + 0.505·10-s − 1.69·11-s + 1.32·13-s − 16-s + 1.19·17-s − 1.20·19-s − 0.357·20-s + 2.40·22-s + 0.380·23-s − 0.871·25-s − 1.87·26-s + 0.0640·29-s + 0.278·31-s + 1.41·32-s − 1.69·34-s − 1.09·37-s + 1.70·38-s − 0.944·41-s + 0.241·43-s − 1.69·44-s − 0.538·46-s + 1.00·47-s + 1.23·50-s + 1.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6182125143\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6182125143\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 10 T + p^{3} T^{2} \) |
| 31 | \( 1 - 48 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 + 248 T + p^{3} T^{2} \) |
| 43 | \( 1 - 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 120 T + p^{3} T^{2} \) |
| 61 | \( 1 + 622 T + p^{3} T^{2} \) |
| 67 | \( 1 - 904 T + p^{3} T^{2} \) |
| 71 | \( 1 - 678 T + p^{3} T^{2} \) |
| 73 | \( 1 - 642 T + p^{3} T^{2} \) |
| 79 | \( 1 - 740 T + p^{3} T^{2} \) |
| 83 | \( 1 - 468 T + p^{3} T^{2} \) |
| 89 | \( 1 - 200 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1266 T + p^{3} T^{2} \) |
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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
−10.58756990037893644291082906655, −9.866244821808345581394716953170, −8.705784094021155244698261931591, −8.128861408024099967876178377529, −7.47016041190547561920798189508, −6.23858665548391049546352502219, −5.01662791105783350033959534517, −3.54261257552292440194803604581, −2.05382218505233693420147085435, −0.61177520299178034000328230038,
0.61177520299178034000328230038, 2.05382218505233693420147085435, 3.54261257552292440194803604581, 5.01662791105783350033959534517, 6.23858665548391049546352502219, 7.47016041190547561920798189508, 8.128861408024099967876178377529, 8.705784094021155244698261931591, 9.866244821808345581394716953170, 10.58756990037893644291082906655