| L(s) = 1 | − 3-s − 5-s − 3·7-s − 2·9-s + 3·11-s + 15-s − 3·17-s + 5·19-s + 3·21-s − 3·23-s + 25-s + 5·27-s + 5·29-s − 3·33-s + 3·35-s − 11·37-s + 5·41-s − 7·43-s + 2·45-s − 8·47-s + 2·49-s + 3·51-s + 10·53-s − 3·55-s − 5·57-s + 59-s + 5·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s − 2/3·9-s + 0.904·11-s + 0.258·15-s − 0.727·17-s + 1.14·19-s + 0.654·21-s − 0.625·23-s + 1/5·25-s + 0.962·27-s + 0.928·29-s − 0.522·33-s + 0.507·35-s − 1.80·37-s + 0.780·41-s − 1.06·43-s + 0.298·45-s − 1.16·47-s + 2/7·49-s + 0.420·51-s + 1.37·53-s − 0.404·55-s − 0.662·57-s + 0.130·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7470658942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7470658942\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.43509614718530, −13.85438502002727, −13.45689074261256, −12.87335788873217, −12.14172700595497, −11.81243930675123, −11.69437175920079, −10.74083164603710, −10.48497880336321, −9.698221845757324, −9.352443191810303, −8.667576500527143, −8.321446828159501, −7.477670769841537, −6.879086858866122, −6.439436346637704, −6.101021528433964, −5.243109549537141, −4.886484149675916, −3.956238704781993, −3.493617812957513, −2.971643466145011, −2.146116040891140, −1.142872400342107, −0.3454972467640066,
0.3454972467640066, 1.142872400342107, 2.146116040891140, 2.971643466145011, 3.493617812957513, 3.956238704781993, 4.886484149675916, 5.243109549537141, 6.101021528433964, 6.439436346637704, 6.879086858866122, 7.477670769841537, 8.321446828159501, 8.667576500527143, 9.352443191810303, 9.698221845757324, 10.48497880336321, 10.74083164603710, 11.69437175920079, 11.81243930675123, 12.14172700595497, 12.87335788873217, 13.45689074261256, 13.85438502002727, 14.43509614718530
