| L(s) = 1 | + 5.28·2-s + 3·3-s + 19.9·4-s + 4.18·5-s + 15.8·6-s − 2.80·7-s + 63.2·8-s + 9·9-s + 22.1·10-s + 3.76·11-s + 59.8·12-s + 43.6·13-s − 14.8·14-s + 12.5·15-s + 174.·16-s + 33.0·17-s + 47.5·18-s + 82.9·19-s + 83.5·20-s − 8.41·21-s + 19.9·22-s − 23·23-s + 189.·24-s − 107.·25-s + 230.·26-s + 27·27-s − 55.9·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 0.577·3-s + 2.49·4-s + 0.374·5-s + 1.07·6-s − 0.151·7-s + 2.79·8-s + 0.333·9-s + 0.699·10-s + 0.103·11-s + 1.44·12-s + 0.931·13-s − 0.283·14-s + 0.216·15-s + 2.73·16-s + 0.472·17-s + 0.623·18-s + 1.00·19-s + 0.933·20-s − 0.0874·21-s + 0.193·22-s − 0.208·23-s + 1.61·24-s − 0.860·25-s + 1.74·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.80155850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.80155850\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 5.28T + 8T^{2} \) |
| 5 | \( 1 - 4.18T + 125T^{2} \) |
| 7 | \( 1 + 2.80T + 343T^{2} \) |
| 11 | \( 1 - 3.76T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.9T + 6.85e3T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 344.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 40.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 430.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 528.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 30.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 85.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 463.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 810.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 343.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 904.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 375.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.66e3T + 9.12e5T^{2} \) |
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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
−8.622047492779382504896788086741, −7.80202988620108932485177713827, −6.89938085487652444521473309823, −6.25676021713651576397059482146, −5.47800704580023605790168773382, −4.75226499207137898466680775169, −3.65772624289812526953457305274, −3.30634702194204291801722716708, −2.22532897372588159299507495003, −1.31846481472096522890637495415,
1.31846481472096522890637495415, 2.22532897372588159299507495003, 3.30634702194204291801722716708, 3.65772624289812526953457305274, 4.75226499207137898466680775169, 5.47800704580023605790168773382, 6.25676021713651576397059482146, 6.89938085487652444521473309823, 7.80202988620108932485177713827, 8.622047492779382504896788086741
