L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 13-s + 15-s + 2·19-s + 9·23-s + 25-s + 27-s − 9·29-s + 8·31-s + 33-s + 2·37-s + 39-s − 3·41-s + 43-s + 45-s + 9·47-s + 9·53-s + 55-s + 2·57-s + 12·59-s + 10·61-s + 65-s + 4·67-s + 9·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s + 0.458·19-s + 1.87·23-s + 1/5·25-s + 0.192·27-s − 1.67·29-s + 1.43·31-s + 0.174·33-s + 0.328·37-s + 0.160·39-s − 0.468·41-s + 0.152·43-s + 0.149·45-s + 1.31·47-s + 1.23·53-s + 0.134·55-s + 0.264·57-s + 1.56·59-s + 1.28·61-s + 0.124·65-s + 0.488·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.374365883\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.374365883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.59304737807335, −13.09477286258503, −12.71671890660398, −12.00874745581371, −11.58371833765248, −11.07014272721910, −10.54971930697444, −10.04398893380215, −9.525625059444662, −9.085043735141945, −8.713311284031772, −8.195033463358290, −7.481479633661687, −7.126556520238384, −6.633904215730206, −5.980640898566268, −5.443545830612935, −4.927792594756209, −4.339615995225918, −3.523303319098657, −3.365067274673430, −2.321187631857858, −2.213990277899541, −1.068367379453954, −0.8075764655677969,
0.8075764655677969, 1.068367379453954, 2.213990277899541, 2.321187631857858, 3.365067274673430, 3.523303319098657, 4.339615995225918, 4.927792594756209, 5.443545830612935, 5.980640898566268, 6.633904215730206, 7.126556520238384, 7.481479633661687, 8.195033463358290, 8.713311284031772, 9.085043735141945, 9.525625059444662, 10.04398893380215, 10.54971930697444, 11.07014272721910, 11.58371833765248, 12.00874745581371, 12.71671890660398, 13.09477286258503, 13.59304737807335