| L(s) = 1 | − 7.82·3-s + 7.04·5-s + 14.1·7-s + 34.2·9-s + 16.9·11-s − 62.0·13-s − 55.1·15-s + 104.·17-s + 9.38·19-s − 110.·21-s + 173.·23-s − 75.4·25-s − 57.1·27-s − 29·29-s − 28.2·31-s − 133.·33-s + 99.6·35-s + 171.·37-s + 485.·39-s + 92.0·41-s + 519.·43-s + 241.·45-s + 109.·47-s − 142.·49-s − 820.·51-s − 140.·53-s + 119.·55-s + ⋯ |
| L(s) = 1 | − 1.50·3-s + 0.629·5-s + 0.763·7-s + 1.27·9-s + 0.465·11-s − 1.32·13-s − 0.949·15-s + 1.49·17-s + 0.113·19-s − 1.15·21-s + 1.57·23-s − 0.603·25-s − 0.407·27-s − 0.185·29-s − 0.163·31-s − 0.702·33-s + 0.481·35-s + 0.762·37-s + 1.99·39-s + 0.350·41-s + 1.84·43-s + 0.800·45-s + 0.340·47-s − 0.416·49-s − 2.25·51-s − 0.364·53-s + 0.293·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.635514246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635514246\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 7.82T + 27T^{2} \) |
| 5 | \( 1 - 7.04T + 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 11 | \( 1 - 16.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.38T + 6.85e3T^{2} \) |
| 23 | \( 1 - 173.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 28.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 171.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 92.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 519.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 109.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 140.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 574.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 535.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 508.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 61.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 826.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 938.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 888.T + 9.12e5T^{2} \) |
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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
−9.161880024289609808234439866647, −7.79677415527092110292813066328, −7.27042560590687566664640331889, −6.30032323580504633682671746924, −5.55229298599754436688436523812, −5.09072890947972885906481939495, −4.26511442615022648066635861166, −2.80769808140121024190190184892, −1.53512241343622635734437426453, −0.69688779632008960473218383317,
0.69688779632008960473218383317, 1.53512241343622635734437426453, 2.80769808140121024190190184892, 4.26511442615022648066635861166, 5.09072890947972885906481939495, 5.55229298599754436688436523812, 6.30032323580504633682671746924, 7.27042560590687566664640331889, 7.79677415527092110292813066328, 9.161880024289609808234439866647
