L(s) = 1 | + 3.46·5-s + 1.73·7-s + 6·11-s + 5.19·13-s − 6·17-s − 5·19-s − 3.46·23-s + 6.99·25-s + 6.92·29-s − 3.46·31-s + 5.99·35-s − 1.73·37-s + 4·43-s + 3.46·47-s − 4·49-s + 6.92·53-s + 20.7·55-s − 6·59-s + 12.1·61-s + 18·65-s + 5·67-s + 13.8·71-s + 7·73-s + 10.3·77-s − 5.19·79-s − 20.7·85-s − 6·89-s + ⋯ |
L(s) = 1 | + 1.54·5-s + 0.654·7-s + 1.80·11-s + 1.44·13-s − 1.45·17-s − 1.14·19-s − 0.722·23-s + 1.39·25-s + 1.28·29-s − 0.622·31-s + 1.01·35-s − 0.284·37-s + 0.609·43-s + 0.505·47-s − 0.571·49-s + 0.951·53-s + 2.80·55-s − 0.781·59-s + 1.55·61-s + 2.23·65-s + 0.610·67-s + 1.64·71-s + 0.819·73-s + 1.18·77-s − 0.584·79-s − 2.25·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.664257368\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.664257368\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.300411797857627049785264583879, −6.88405827855310625991161834855, −6.45065265112369607571706103496, −6.08510754154932167193405460587, −5.20972087200152945944826200360, −4.26049919965862245325916036893, −3.80328492787893421372814205960, −2.39886901292792307315453109541, −1.80938351194483891444164145221, −1.07841686164402336073494784918,
1.07841686164402336073494784918, 1.80938351194483891444164145221, 2.39886901292792307315453109541, 3.80328492787893421372814205960, 4.26049919965862245325916036893, 5.20972087200152945944826200360, 6.08510754154932167193405460587, 6.45065265112369607571706103496, 6.88405827855310625991161834855, 8.300411797857627049785264583879