| L(s) = 1 | + 3-s − 5-s + 3.74·7-s + 9-s − 0.896·11-s − 1.29·13-s − 15-s − 4.92·17-s − 3.04·19-s + 3.74·21-s + 1.90·23-s + 25-s + 27-s + 8.22·29-s + 31-s − 0.896·33-s − 3.74·35-s + 8.78·37-s − 1.29·39-s + 10.0·41-s + 5.02·43-s − 45-s − 1.00·47-s + 7.03·49-s − 4.92·51-s − 2.01·53-s + 0.896·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.41·7-s + 0.333·9-s − 0.270·11-s − 0.358·13-s − 0.258·15-s − 1.19·17-s − 0.697·19-s + 0.817·21-s + 0.397·23-s + 0.200·25-s + 0.192·27-s + 1.52·29-s + 0.179·31-s − 0.155·33-s − 0.633·35-s + 1.44·37-s − 0.207·39-s + 1.57·41-s + 0.766·43-s − 0.149·45-s − 0.147·47-s + 1.00·49-s − 0.690·51-s − 0.276·53-s + 0.120·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.495076398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.495076398\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + 0.896T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 - 1.90T + 23T^{2} \) |
| 29 | \( 1 - 8.22T + 29T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.02T + 43T^{2} \) |
| 47 | \( 1 + 1.00T + 47T^{2} \) |
| 53 | \( 1 + 2.01T + 53T^{2} \) |
| 59 | \( 1 - 2.28T + 59T^{2} \) |
| 61 | \( 1 - 9.93T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 2.16T + 71T^{2} \) |
| 73 | \( 1 - 4.17T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−8.299399785287788509162343445514, −8.039752304171608038799031115536, −7.18759477353026555578989574195, −6.45438845041403778034984426111, −5.33946477377461025626387228120, −4.45544771438207239969943564905, −4.20749040002192000155008074388, −2.76151941451924083031857911591, −2.17489107968255585438146228999, −0.917690129591161090674844710887,
0.917690129591161090674844710887, 2.17489107968255585438146228999, 2.76151941451924083031857911591, 4.20749040002192000155008074388, 4.45544771438207239969943564905, 5.33946477377461025626387228120, 6.45438845041403778034984426111, 7.18759477353026555578989574195, 8.039752304171608038799031115536, 8.299399785287788509162343445514
