| L(s) = 1 | − 2.32·3-s + 5-s − 1.33·7-s + 2.42·9-s − 5.57·11-s − 5.93·13-s − 2.32·15-s + 3.42·17-s − 1.70·19-s + 3.11·21-s − 6.19·23-s + 25-s + 1.33·27-s − 2.54·29-s − 1.32·31-s + 12.9·33-s − 1.33·35-s − 5.25·37-s + 13.8·39-s − 6.82·41-s − 8.53·43-s + 2.42·45-s + 2.81·47-s − 5.21·49-s − 7.96·51-s − 4.55·53-s − 5.57·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s − 0.504·7-s + 0.809·9-s − 1.68·11-s − 1.64·13-s − 0.601·15-s + 0.829·17-s − 0.390·19-s + 0.679·21-s − 1.29·23-s + 0.200·25-s + 0.256·27-s − 0.472·29-s − 0.238·31-s + 2.26·33-s − 0.225·35-s − 0.863·37-s + 2.21·39-s − 1.06·41-s − 1.30·43-s + 0.361·45-s + 0.411·47-s − 0.745·49-s − 1.11·51-s − 0.625·53-s − 0.752·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06871433059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06871433059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 5.93T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 + 6.19T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 + 8.53T + 43T^{2} \) |
| 47 | \( 1 - 2.81T + 47T^{2} \) |
| 53 | \( 1 + 4.55T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 - 6.36T + 73T^{2} \) |
| 79 | \( 1 + 9.96T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.72351327826145403221747198975, −7.03071748957761383969024970304, −6.32367045810200198141731588681, −5.67191012594796397222963264166, −5.10687156084109155728416999884, −4.75082971847936690106827882355, −3.45367925757441397345354476402, −2.60706192857464209146188951831, −1.74540868358577934087202360147, −0.13107729578934245777685665391,
0.13107729578934245777685665391, 1.74540868358577934087202360147, 2.60706192857464209146188951831, 3.45367925757441397345354476402, 4.75082971847936690106827882355, 5.10687156084109155728416999884, 5.67191012594796397222963264166, 6.32367045810200198141731588681, 7.03071748957761383969024970304, 7.72351327826145403221747198975
