| L(s) = 1 | − 1.07·3-s + 5-s − 3.77·7-s − 1.83·9-s − 0.592·11-s − 3.63·13-s − 1.07·15-s + 4.24·17-s − 2.63·19-s + 4.07·21-s − 4.35·23-s + 25-s + 5.21·27-s + 0.768·29-s − 6.69·31-s + 0.639·33-s − 3.77·35-s − 37-s + 3.92·39-s + 7.71·41-s − 2.33·43-s − 1.83·45-s + 7.64·47-s + 7.22·49-s − 4.58·51-s + 0.432·53-s − 0.592·55-s + ⋯ |
| L(s) = 1 | − 0.623·3-s + 0.447·5-s − 1.42·7-s − 0.611·9-s − 0.178·11-s − 1.00·13-s − 0.278·15-s + 1.02·17-s − 0.605·19-s + 0.888·21-s − 0.908·23-s + 0.200·25-s + 1.00·27-s + 0.142·29-s − 1.20·31-s + 0.111·33-s − 0.637·35-s − 0.164·37-s + 0.628·39-s + 1.20·41-s − 0.355·43-s − 0.273·45-s + 1.11·47-s + 1.03·49-s − 0.641·51-s + 0.0593·53-s − 0.0798·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7542348610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7542348610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 1.07T + 3T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + 0.592T + 11T^{2} \) |
| 13 | \( 1 + 3.63T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 2.63T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 - 0.768T + 29T^{2} \) |
| 31 | \( 1 + 6.69T + 31T^{2} \) |
| 41 | \( 1 - 7.71T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 - 0.432T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.39T + 61T^{2} \) |
| 67 | \( 1 + 4.83T + 67T^{2} \) |
| 71 | \( 1 + 0.639T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 7.77T + 97T^{2} \) |
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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
−8.961901378353080485642396292709, −7.895033343319986675895525431153, −7.11563452314868018275432461973, −6.30087486793632334366864497153, −5.77671344164876662033894535441, −5.13578020468947597300714635847, −3.95694358817503896509295919699, −3.03303487246930554193912499243, −2.21225707187621860487115311331, −0.51551098949183428264325388381,
0.51551098949183428264325388381, 2.21225707187621860487115311331, 3.03303487246930554193912499243, 3.95694358817503896509295919699, 5.13578020468947597300714635847, 5.77671344164876662033894535441, 6.30087486793632334366864497153, 7.11563452314868018275432461973, 7.895033343319986675895525431153, 8.961901378353080485642396292709
