L(s) = 1 | + 1.87·2-s + 0.652·3-s + 1.53·4-s + 3.53·5-s + 1.22·6-s − 0.879·8-s − 2.57·9-s + 6.63·10-s + 11-s + 12-s + 4.41·13-s + 2.30·15-s − 4.71·16-s − 5.24·17-s − 4.83·18-s + 1.81·19-s + 5.41·20-s + 1.87·22-s − 6.33·23-s − 0.573·24-s + 7.47·25-s + 8.29·26-s − 3.63·27-s + 1.92·29-s + 4.33·30-s + 1.46·31-s − 7.10·32-s + ⋯ |
L(s) = 1 | + 1.32·2-s + 0.376·3-s + 0.766·4-s + 1.57·5-s + 0.500·6-s − 0.310·8-s − 0.857·9-s + 2.09·10-s + 0.301·11-s + 0.288·12-s + 1.22·13-s + 0.595·15-s − 1.17·16-s − 1.27·17-s − 1.14·18-s + 0.416·19-s + 1.21·20-s + 0.400·22-s − 1.32·23-s − 0.117·24-s + 1.49·25-s + 1.62·26-s − 0.700·27-s + 0.356·29-s + 0.791·30-s + 0.263·31-s − 1.25·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.558364933\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.558364933\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 3 | \( 1 - 0.652T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 - 0.283T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 + 4.55T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 9.53T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 + 0.694T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−11.07610577684918782863648438307, −9.884353127172674007247948200321, −9.070234680289049851660992534269, −8.345481287677819273620689181043, −6.53755656665418312317099221670, −6.09708071507662225365662524352, −5.31262599921936768386931761341, −4.13196420143855307039189995532, −2.98306147523787437364138532829, −1.98196403479871147831307618055,
1.98196403479871147831307618055, 2.98306147523787437364138532829, 4.13196420143855307039189995532, 5.31262599921936768386931761341, 6.09708071507662225365662524352, 6.53755656665418312317099221670, 8.345481287677819273620689181043, 9.070234680289049851660992534269, 9.884353127172674007247948200321, 11.07610577684918782863648438307