| L(s) = 1 | + 0.536·2-s − 1.71·4-s + 1.82·5-s + 2.96·7-s − 1.99·8-s + 0.980·10-s − 11-s + 4.10·13-s + 1.58·14-s + 2.35·16-s − 0.658·17-s − 4.67·19-s − 3.12·20-s − 0.536·22-s + 1.60·23-s − 1.66·25-s + 2.20·26-s − 5.06·28-s + 6.71·29-s − 3.25·31-s + 5.24·32-s − 0.353·34-s + 5.40·35-s + 3.16·37-s − 2.51·38-s − 3.64·40-s + 9.51·41-s + ⋯ |
| L(s) = 1 | + 0.379·2-s − 0.855·4-s + 0.817·5-s + 1.11·7-s − 0.704·8-s + 0.310·10-s − 0.301·11-s + 1.13·13-s + 0.424·14-s + 0.588·16-s − 0.159·17-s − 1.07·19-s − 0.699·20-s − 0.114·22-s + 0.334·23-s − 0.332·25-s + 0.432·26-s − 0.957·28-s + 1.24·29-s − 0.583·31-s + 0.927·32-s − 0.0606·34-s + 0.914·35-s + 0.520·37-s − 0.407·38-s − 0.575·40-s + 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658008457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658008457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.536T + 2T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 + 0.658T + 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 - 6.71T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 - 9.51T + 41T^{2} \) |
| 43 | \( 1 - 7.90T + 43T^{2} \) |
| 47 | \( 1 - 3.05T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 - 7.97T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.145580950893929019034379525692, −7.54788168282943485609544086471, −6.19000376938923047040757548020, −6.04242365140902134584643511898, −5.09583347247351534785253693061, −4.51729158873922087204776755799, −3.87379189358416475323951422573, −2.77786391725820688430770479847, −1.84646426276374854808590303931, −0.850328148666634533585686300139,
0.850328148666634533585686300139, 1.84646426276374854808590303931, 2.77786391725820688430770479847, 3.87379189358416475323951422573, 4.51729158873922087204776755799, 5.09583347247351534785253693061, 6.04242365140902134584643511898, 6.19000376938923047040757548020, 7.54788168282943485609544086471, 8.145580950893929019034379525692
