| L(s) = 1 | − 2.62·3-s + 5-s − 4.07·7-s + 3.88·9-s + 1.05·11-s − 4.00·13-s − 2.62·15-s − 5.19·17-s − 1.19·19-s + 10.7·21-s + 2.46·23-s + 25-s − 2.32·27-s + 3.14·29-s + 11.0·31-s − 2.77·33-s − 4.07·35-s − 0.856·37-s + 10.4·39-s − 8.16·41-s − 2.00·43-s + 3.88·45-s + 8.74·47-s + 9.62·49-s + 13.6·51-s − 7.04·53-s + 1.05·55-s + ⋯ |
| L(s) = 1 | − 1.51·3-s + 0.447·5-s − 1.54·7-s + 1.29·9-s + 0.318·11-s − 1.10·13-s − 0.677·15-s − 1.26·17-s − 0.274·19-s + 2.33·21-s + 0.513·23-s + 0.200·25-s − 0.448·27-s + 0.583·29-s + 1.98·31-s − 0.482·33-s − 0.689·35-s − 0.140·37-s + 1.68·39-s − 1.27·41-s − 0.305·43-s + 0.579·45-s + 1.27·47-s + 1.37·49-s + 1.91·51-s − 0.967·53-s + 0.142·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.62T + 3T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 + 4.00T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 + 1.19T + 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 - 11.0T + 31T^{2} \) |
| 37 | \( 1 + 0.856T + 37T^{2} \) |
| 41 | \( 1 + 8.16T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 + 0.749T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 1.29T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + 1.60T + 73T^{2} \) |
| 79 | \( 1 - 8.50T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−6.93942629001642655897334142807, −6.63382247825354159884166998525, −6.33979805492522313478955174551, −5.42217486384636188292247145787, −4.84059577759367336066486940055, −4.13501974142789345978165377663, −3.02084300193520735572254768382, −2.26159395244336232880268428273, −0.868943476824904754733809242946, 0,
0.868943476824904754733809242946, 2.26159395244336232880268428273, 3.02084300193520735572254768382, 4.13501974142789345978165377663, 4.84059577759367336066486940055, 5.42217486384636188292247145787, 6.33979805492522313478955174551, 6.63382247825354159884166998525, 6.93942629001642655897334142807
