| L(s) = 1 | + 0.220·3-s + 5·5-s + 25.5·7-s − 26.9·9-s + 16.2·11-s − 78.1·13-s + 1.10·15-s − 58.2·17-s + 144.·19-s + 5.64·21-s − 38.1·23-s + 25·25-s − 11.9·27-s + 29·29-s + 343.·31-s + 3.58·33-s + 127.·35-s + 180.·37-s − 17.2·39-s − 189.·41-s + 336.·43-s − 134.·45-s + 134.·47-s + 310.·49-s − 12.8·51-s − 285.·53-s + 81.2·55-s + ⋯ |
| L(s) = 1 | + 0.0425·3-s + 0.447·5-s + 1.38·7-s − 0.998·9-s + 0.445·11-s − 1.66·13-s + 0.0190·15-s − 0.831·17-s + 1.74·19-s + 0.0586·21-s − 0.346·23-s + 0.200·25-s − 0.0849·27-s + 0.185·29-s + 1.98·31-s + 0.0189·33-s + 0.617·35-s + 0.803·37-s − 0.0708·39-s − 0.721·41-s + 1.19·43-s − 0.446·45-s + 0.416·47-s + 0.906·49-s − 0.0353·51-s − 0.740·53-s + 0.199·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.517459846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517459846\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 0.220T + 27T^{2} \) |
| 7 | \( 1 - 25.5T + 343T^{2} \) |
| 11 | \( 1 - 16.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 38.1T + 1.21e4T^{2} \) |
| 31 | \( 1 - 343.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 180.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 189.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 97.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 933.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 55.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 548.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 249.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 682.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 374.T + 9.12e5T^{2} \) |
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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
−9.453561280040872681913989942272, −8.537984318645135660765675728374, −7.84334037149292055922197033078, −7.02779490640224748003848571694, −5.90960777951072211104696129272, −5.07377988631770531744967044088, −4.46795408000638367284312189281, −2.90745181980674411364064395524, −2.11873182426951747903387552417, −0.826485731639871749261115427052,
0.826485731639871749261115427052, 2.11873182426951747903387552417, 2.90745181980674411364064395524, 4.46795408000638367284312189281, 5.07377988631770531744967044088, 5.90960777951072211104696129272, 7.02779490640224748003848571694, 7.84334037149292055922197033078, 8.537984318645135660765675728374, 9.453561280040872681913989942272
