| L(s) = 1 | − 2.78·3-s − 5-s − 0.646·7-s + 4.73·9-s + 2.99·11-s − 6.15·13-s + 2.78·15-s + 2.72·17-s + 19-s + 1.79·21-s − 0.646·23-s + 25-s − 4.82·27-s − 3.81·29-s + 0.0145·31-s − 8.32·33-s + 0.646·35-s − 6.49·37-s + 17.1·39-s + 0.532·41-s + 2.45·43-s − 4.73·45-s + 11.8·47-s − 6.58·49-s − 7.58·51-s + 1.29·53-s − 2.99·55-s + ⋯ |
| L(s) = 1 | − 1.60·3-s − 0.447·5-s − 0.244·7-s + 1.57·9-s + 0.902·11-s − 1.70·13-s + 0.718·15-s + 0.661·17-s + 0.229·19-s + 0.392·21-s − 0.134·23-s + 0.200·25-s − 0.927·27-s − 0.708·29-s + 0.00261·31-s − 1.44·33-s + 0.109·35-s − 1.06·37-s + 2.74·39-s + 0.0832·41-s + 0.375·43-s − 0.705·45-s + 1.73·47-s − 0.940·49-s − 1.06·51-s + 0.178·53-s − 0.403·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5884962540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5884962540\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.78T + 3T^{2} \) |
| 7 | \( 1 + 0.646T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 + 6.15T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 23 | \( 1 + 0.646T + 23T^{2} \) |
| 29 | \( 1 + 3.81T + 29T^{2} \) |
| 31 | \( 1 - 0.0145T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 - 0.532T + 41T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 1.29T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 3.75T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 1.46T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 6.34T + 97T^{2} \) |
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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
−7.78499974620292926904780472257, −7.09784583590092447926791774758, −6.78913535676211702593988495313, −5.72907045453307722661691275408, −5.40304837439499034771571378062, −4.52074126137808369114925983825, −3.92691771031397889755977224811, −2.80825982562050162263669153413, −1.53771783217446768693944718964, −0.45131915795852273483406983696,
0.45131915795852273483406983696, 1.53771783217446768693944718964, 2.80825982562050162263669153413, 3.92691771031397889755977224811, 4.52074126137808369114925983825, 5.40304837439499034771571378062, 5.72907045453307722661691275408, 6.78913535676211702593988495313, 7.09784583590092447926791774758, 7.78499974620292926904780472257
