| L(s) = 1 | + 2-s + 0.698·3-s + 4-s − 4.21·5-s + 0.698·6-s − 4.34·7-s + 8-s − 2.51·9-s − 4.21·10-s + 0.125·11-s + 0.698·12-s + 13-s − 4.34·14-s − 2.94·15-s + 16-s + 5.01·17-s − 2.51·18-s − 2.67·19-s − 4.21·20-s − 3.03·21-s + 0.125·22-s + 6.96·23-s + 0.698·24-s + 12.7·25-s + 26-s − 3.85·27-s − 4.34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.403·3-s + 0.5·4-s − 1.88·5-s + 0.285·6-s − 1.64·7-s + 0.353·8-s − 0.837·9-s − 1.33·10-s + 0.0378·11-s + 0.201·12-s + 0.277·13-s − 1.16·14-s − 0.760·15-s + 0.250·16-s + 1.21·17-s − 0.592·18-s − 0.613·19-s − 0.942·20-s − 0.662·21-s + 0.0267·22-s + 1.45·23-s + 0.142·24-s + 2.55·25-s + 0.196·26-s − 0.740·27-s − 0.821·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455580296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455580296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 3 | \( 1 - 0.698T + 3T^{2} \) |
| 5 | \( 1 + 4.21T + 5T^{2} \) |
| 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 - 0.125T + 11T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 + 2.67T + 19T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 - 3.22T + 41T^{2} \) |
| 43 | \( 1 - 9.69T + 43T^{2} \) |
| 47 | \( 1 - 8.90T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 1.93T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 1.30T + 71T^{2} \) |
| 73 | \( 1 - 4.08T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−8.860275332199995730828435154477, −7.88734482583200048230198594826, −7.37687710823220824666992317372, −6.57553220142581667459436484704, −5.79507133096305335512707598035, −4.75230735853921757446364495875, −3.71496946516955800758827266688, −3.36963736813937954031457626388, −2.74241185520310464938896667702, −0.63687273625465252682076377040,
0.63687273625465252682076377040, 2.74241185520310464938896667702, 3.36963736813937954031457626388, 3.71496946516955800758827266688, 4.75230735853921757446364495875, 5.79507133096305335512707598035, 6.57553220142581667459436484704, 7.37687710823220824666992317372, 7.88734482583200048230198594826, 8.860275332199995730828435154477
