| L(s) = 1 | + 0.238·3-s − 5·5-s + 7·7-s − 26.9·9-s + 54.5·11-s − 52.2·13-s − 1.19·15-s + 49.8·17-s − 161.·19-s + 1.67·21-s − 119.·23-s + 25·25-s − 12.8·27-s + 49.5·29-s − 164.·31-s + 13.0·33-s − 35·35-s − 216.·37-s − 12.4·39-s − 252.·41-s − 68.7·43-s + 134.·45-s − 84.4·47-s + 49·49-s + 11.9·51-s − 204.·53-s − 272.·55-s + ⋯ |
| L(s) = 1 | + 0.0459·3-s − 0.447·5-s + 0.377·7-s − 0.997·9-s + 1.49·11-s − 1.11·13-s − 0.0205·15-s + 0.711·17-s − 1.94·19-s + 0.0173·21-s − 1.08·23-s + 0.200·25-s − 0.0917·27-s + 0.317·29-s − 0.955·31-s + 0.0687·33-s − 0.169·35-s − 0.963·37-s − 0.0511·39-s − 0.962·41-s − 0.243·43-s + 0.446·45-s − 0.261·47-s + 0.142·49-s + 0.0327·51-s − 0.530·53-s − 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 0.238T + 27T^{2} \) |
| 11 | \( 1 - 54.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 161.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 252.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 68.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + 84.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 204.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 552.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 178.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 415.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 931.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 223.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.70e3T + 9.12e5T^{2} \) |
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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
−11.09400285730912064510012968337, −10.01365456843684949093161602689, −8.877871900178125958028739038189, −8.185163510452554323244366706898, −6.99253173304800319917357416377, −5.95824328288277188531800759001, −4.64392173634943651412979403653, −3.53677550713164465680795550794, −1.95886763850354033845189511793, 0,
1.95886763850354033845189511793, 3.53677550713164465680795550794, 4.64392173634943651412979403653, 5.95824328288277188531800759001, 6.99253173304800319917357416377, 8.185163510452554323244366706898, 8.877871900178125958028739038189, 10.01365456843684949093161602689, 11.09400285730912064510012968337
