| L(s) = 1 | − 2.91·2-s − 3·3-s + 0.525·4-s + 16.8·5-s + 8.75·6-s + 26.0·7-s + 21.8·8-s + 9·9-s − 49.1·10-s + 11·11-s − 1.57·12-s + 31.7·13-s − 76.0·14-s − 50.4·15-s − 67.9·16-s + 83.9·17-s − 26.2·18-s − 2.09·19-s + 8.84·20-s − 78.1·21-s − 32.1·22-s + 114.·23-s − 65.4·24-s + 158.·25-s − 92.7·26-s − 27·27-s + 13.6·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 0.577·3-s + 0.0656·4-s + 1.50·5-s + 0.596·6-s + 1.40·7-s + 0.964·8-s + 0.333·9-s − 1.55·10-s + 0.301·11-s − 0.0379·12-s + 0.677·13-s − 1.45·14-s − 0.868·15-s − 1.06·16-s + 1.19·17-s − 0.344·18-s − 0.0253·19-s + 0.0988·20-s − 0.812·21-s − 0.311·22-s + 1.04·23-s − 0.556·24-s + 1.26·25-s − 0.699·26-s − 0.192·27-s + 0.0924·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.106109391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106109391\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 2.91T + 8T^{2} \) |
| 5 | \( 1 - 16.8T + 125T^{2} \) |
| 7 | \( 1 - 26.0T + 343T^{2} \) |
| 13 | \( 1 - 31.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 83.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.09T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 15.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 54.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 755.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 196.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 90.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 186.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 749.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−9.033301549306765324010858755633, −8.088358475220029140305229927106, −7.45235208622155077892372828666, −6.43829747276334103875681647719, −5.53558948510832907913241390274, −5.05908215610492888179073306920, −4.02925902552248314051185913973, −2.36189442737201428516522740524, −1.34446730625601416467977863304, −1.00175957244989384616144159222,
1.00175957244989384616144159222, 1.34446730625601416467977863304, 2.36189442737201428516522740524, 4.02925902552248314051185913973, 5.05908215610492888179073306920, 5.53558948510832907913241390274, 6.43829747276334103875681647719, 7.45235208622155077892372828666, 8.088358475220029140305229927106, 9.033301549306765324010858755633
