| L(s) = 1 | − 3.28·3-s + 5-s + 4.04·7-s + 7.81·9-s − 2.68·11-s − 5.26·13-s − 3.28·15-s − 2.40·17-s − 3.25·19-s − 13.3·21-s − 6.20·23-s + 25-s − 15.8·27-s − 2.35·29-s + 2.69·31-s + 8.84·33-s + 4.04·35-s + 7.59·37-s + 17.3·39-s − 1.93·41-s − 11.7·43-s + 7.81·45-s + 10.3·47-s + 9.37·49-s + 7.91·51-s + 8.63·53-s − 2.68·55-s + ⋯ |
| L(s) = 1 | − 1.89·3-s + 0.447·5-s + 1.52·7-s + 2.60·9-s − 0.810·11-s − 1.45·13-s − 0.849·15-s − 0.583·17-s − 0.745·19-s − 2.90·21-s − 1.29·23-s + 0.200·25-s − 3.04·27-s − 0.436·29-s + 0.483·31-s + 1.53·33-s + 0.684·35-s + 1.24·37-s + 2.77·39-s − 0.301·41-s − 1.78·43-s + 1.16·45-s + 1.51·47-s + 1.33·49-s + 1.10·51-s + 1.18·53-s − 0.362·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8248291341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8248291341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 3.28T + 3T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 + 2.40T + 17T^{2} \) |
| 19 | \( 1 + 3.25T + 19T^{2} \) |
| 23 | \( 1 + 6.20T + 23T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 7.59T + 37T^{2} \) |
| 41 | \( 1 + 1.93T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 8.63T + 53T^{2} \) |
| 59 | \( 1 - 6.52T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 + 8.22T + 67T^{2} \) |
| 71 | \( 1 + 6.95T + 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 + 9.84T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−7.85440201492726307859613178645, −7.27192068800837892390534202235, −6.57015054990854986322535833215, −5.68767350686107038870165184567, −5.33615063889220281480897271739, −4.58365590902259542679883997235, −4.25964702400457382406489659446, −2.35062976585294341996026893872, −1.77500849518295772860594606166, −0.52072337792663608920185097669,
0.52072337792663608920185097669, 1.77500849518295772860594606166, 2.35062976585294341996026893872, 4.25964702400457382406489659446, 4.58365590902259542679883997235, 5.33615063889220281480897271739, 5.68767350686107038870165184567, 6.57015054990854986322535833215, 7.27192068800837892390534202235, 7.85440201492726307859613178645
