| L(s) = 1 | + 3-s − 5-s − 0.212·7-s + 9-s − 2.52·11-s − 3.33·13-s − 15-s − 7.70·17-s + 7.01·19-s − 0.212·21-s − 1.74·23-s + 25-s + 27-s + 1.69·29-s − 2.26·31-s − 2.52·33-s + 0.212·35-s + 9.86·37-s − 3.33·39-s + 4.26·41-s − 1.58·43-s − 45-s − 5.44·47-s − 6.95·49-s − 7.70·51-s + 6.91·53-s + 2.52·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.0802·7-s + 0.333·9-s − 0.762·11-s − 0.923·13-s − 0.258·15-s − 1.86·17-s + 1.60·19-s − 0.0463·21-s − 0.363·23-s + 0.200·25-s + 0.192·27-s + 0.314·29-s − 0.407·31-s − 0.440·33-s + 0.0359·35-s + 1.62·37-s − 0.533·39-s + 0.666·41-s − 0.242·43-s − 0.149·45-s − 0.793·47-s − 0.993·49-s − 1.07·51-s + 0.949·53-s + 0.341·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.657625693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.657625693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 0.212T + 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 + 2.26T + 31T^{2} \) |
| 37 | \( 1 - 9.86T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 + 5.44T + 47T^{2} \) |
| 53 | \( 1 - 6.91T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 - 8.27T + 61T^{2} \) |
| 71 | \( 1 - 4.11T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 - 6.07T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 7.28T + 97T^{2} \) |
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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
−7.83571642177540022026815082303, −7.26156377893633765481780814573, −6.64931900407402984229321026781, −5.70244773187608154477416725364, −4.83591989818183792963083344930, −4.38134817709042252580172667534, −3.39056787515570647044632171492, −2.68072120223795021521353047163, −2.00053063930800597301056121365, −0.59250365428848048760976506317,
0.59250365428848048760976506317, 2.00053063930800597301056121365, 2.68072120223795021521353047163, 3.39056787515570647044632171492, 4.38134817709042252580172667534, 4.83591989818183792963083344930, 5.70244773187608154477416725364, 6.64931900407402984229321026781, 7.26156377893633765481780814573, 7.83571642177540022026815082303
