| L(s) = 1 | + 0.618·3-s − 5-s − 2.85·7-s − 2.61·9-s − 3.23·11-s − 5.09·13-s − 0.618·15-s − 3.61·17-s + 2.76·19-s − 1.76·21-s + 5.85·23-s + 25-s − 3.47·27-s + 29-s − 1.61·31-s − 2.00·33-s + 2.85·35-s + 9.23·37-s − 3.14·39-s − 3.70·41-s − 7.61·43-s + 2.61·45-s − 8·47-s + 1.14·49-s − 2.23·51-s + 9.32·53-s + 3.23·55-s + ⋯ |
| L(s) = 1 | + 0.356·3-s − 0.447·5-s − 1.07·7-s − 0.872·9-s − 0.975·11-s − 1.41·13-s − 0.159·15-s − 0.877·17-s + 0.634·19-s − 0.384·21-s + 1.22·23-s + 0.200·25-s − 0.668·27-s + 0.185·29-s − 0.290·31-s − 0.348·33-s + 0.482·35-s + 1.51·37-s − 0.503·39-s − 0.579·41-s − 1.16·43-s + 0.390·45-s − 1.16·47-s + 0.163·49-s − 0.313·51-s + 1.28·53-s + 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7205198556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7205198556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 + 3.23T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 31 | \( 1 + 1.61T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 + 7.61T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 9.32T + 53T^{2} \) |
| 59 | \( 1 + 9.38T + 59T^{2} \) |
| 61 | \( 1 - 2.14T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.90T + 73T^{2} \) |
| 79 | \( 1 + 2.90T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 8.79T + 97T^{2} \) |
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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
−8.257909003371320681937299431989, −7.61899310432646710402783165390, −6.94537000473268447033706961348, −6.23394395621112211322025120020, −5.21350721456265352209923151200, −4.74865931687511623892018378853, −3.48608530734305423702668303049, −2.92237715625787114604272229517, −2.27875871819941325928168641245, −0.42649675330621668308101257907,
0.42649675330621668308101257907, 2.27875871819941325928168641245, 2.92237715625787114604272229517, 3.48608530734305423702668303049, 4.74865931687511623892018378853, 5.21350721456265352209923151200, 6.23394395621112211322025120020, 6.94537000473268447033706961348, 7.61899310432646710402783165390, 8.257909003371320681937299431989
