| L(s) = 1 | − 5.03·2-s + 3·3-s + 17.3·4-s + 13.6·5-s − 15.1·6-s + 24.5·7-s − 47.1·8-s + 9·9-s − 68.5·10-s + 69.5·11-s + 52.0·12-s + 28.1·13-s − 123.·14-s + 40.8·15-s + 98.4·16-s + 5.33·17-s − 45.3·18-s − 23.0·19-s + 236.·20-s + 73.6·21-s − 350.·22-s + 23·23-s − 141.·24-s + 60.3·25-s − 141.·26-s + 27·27-s + 426.·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.16·4-s + 1.21·5-s − 1.02·6-s + 1.32·7-s − 2.08·8-s + 0.333·9-s − 2.16·10-s + 1.90·11-s + 1.25·12-s + 0.601·13-s − 2.36·14-s + 0.702·15-s + 1.53·16-s + 0.0760·17-s − 0.593·18-s − 0.278·19-s + 2.64·20-s + 0.765·21-s − 3.39·22-s + 0.208·23-s − 1.20·24-s + 0.482·25-s − 1.07·26-s + 0.192·27-s + 2.87·28-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
| \(L(2)\) |
\(\approx\) |
\(2.419155202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.419155202\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.03T + 8T^{2} \) |
| 5 | \( 1 - 13.6T + 125T^{2} \) |
| 7 | \( 1 - 24.5T + 343T^{2} \) |
| 11 | \( 1 - 69.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.33T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.0T + 6.85e3T^{2} \) |
| 31 | \( 1 + 55.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 131.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 87.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 82.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 443.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 607.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 17.5T + 2.05e5T^{2} \) |
| 61 | \( 1 - 47.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 743.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 418.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 166.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 529.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 560.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3T + 9.12e5T^{2} \) |
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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
−8.820416502869974303088826903623, −8.335036095714015869968853142978, −7.54094770609540374794949879962, −6.60096151296691336177610610840, −6.11209296523876836876803502325, −4.80861609604864588692623340981, −3.56495919552558080727065603783, −2.16259990722298603484772660522, −1.63078634548684567120982621713, −1.03176266350856376557818487830,
1.03176266350856376557818487830, 1.63078634548684567120982621713, 2.16259990722298603484772660522, 3.56495919552558080727065603783, 4.80861609604864588692623340981, 6.11209296523876836876803502325, 6.60096151296691336177610610840, 7.54094770609540374794949879962, 8.335036095714015869968853142978, 8.820416502869974303088826903623
