| L(s) = 1 | + 3.12·3-s + 3.59·5-s − 1.40·7-s + 6.75·9-s − 1.93·11-s − 3.03·13-s + 11.2·15-s + 1.26·17-s + 7.89·19-s − 4.38·21-s + 5.44·23-s + 7.91·25-s + 11.7·27-s − 7.16·29-s − 0.730·31-s − 6.05·33-s − 5.04·35-s − 10.6·37-s − 9.49·39-s − 5.78·41-s − 43-s + 24.2·45-s + 2.66·47-s − 5.02·49-s + 3.96·51-s + 9.42·53-s − 6.96·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 1.60·5-s − 0.530·7-s + 2.25·9-s − 0.584·11-s − 0.842·13-s + 2.89·15-s + 0.307·17-s + 1.81·19-s − 0.957·21-s + 1.13·23-s + 1.58·25-s + 2.25·27-s − 1.33·29-s − 0.131·31-s − 1.05·33-s − 0.853·35-s − 1.74·37-s − 1.51·39-s − 0.904·41-s − 0.152·43-s + 3.62·45-s + 0.389·47-s − 0.718·49-s + 0.555·51-s + 1.29·53-s − 0.939·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.543120777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.543120777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 + 1.40T + 7T^{2} \) |
| 11 | \( 1 + 1.93T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 7.89T + 19T^{2} \) |
| 23 | \( 1 - 5.44T + 23T^{2} \) |
| 29 | \( 1 + 7.16T + 29T^{2} \) |
| 31 | \( 1 + 0.730T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 5.78T + 41T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 0.102T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 6.98T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−9.083350289108808218125787398240, −8.184949372082911683612745374228, −7.21606367279924134492997240919, −6.94227299797485822482648544956, −5.50741259357331475430097943879, −5.11196047465448136474748284470, −3.63048754988971601924021402565, −2.96570625975012884438643421543, −2.29416089576172708728299517446, −1.41755536919377481947203567501,
1.41755536919377481947203567501, 2.29416089576172708728299517446, 2.96570625975012884438643421543, 3.63048754988971601924021402565, 5.11196047465448136474748284470, 5.50741259357331475430097943879, 6.94227299797485822482648544956, 7.21606367279924134492997240919, 8.184949372082911683612745374228, 9.083350289108808218125787398240
