| L(s) = 1 | + 2-s − 3-s + 4-s + 3.17·5-s − 6-s + 0.381·7-s + 8-s + 9-s + 3.17·10-s + 2.66·11-s − 12-s − 7.04·13-s + 0.381·14-s − 3.17·15-s + 16-s + 4.68·17-s + 18-s + 3.17·20-s − 0.381·21-s + 2.66·22-s + 1.43·23-s − 24-s + 5.08·25-s − 7.04·26-s − 27-s + 0.381·28-s + 8.20·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.42·5-s − 0.408·6-s + 0.144·7-s + 0.353·8-s + 0.333·9-s + 1.00·10-s + 0.803·11-s − 0.288·12-s − 1.95·13-s + 0.102·14-s − 0.819·15-s + 0.250·16-s + 1.13·17-s + 0.235·18-s + 0.710·20-s − 0.0833·21-s + 0.568·22-s + 0.298·23-s − 0.204·24-s + 1.01·25-s − 1.38·26-s − 0.192·27-s + 0.0721·28-s + 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.186278435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.186278435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 13 | \( 1 + 7.04T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 - 8.62T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 3.25T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 + 8.20T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 3.84T + 73T^{2} \) |
| 79 | \( 1 - 1.39T + 79T^{2} \) |
| 83 | \( 1 + 4.31T + 83T^{2} \) |
| 89 | \( 1 + 7.80T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.387240155422170765654729871172, −8.201977283420626733633363323149, −7.18289590960962294788035624583, −6.52821472562418635970661971261, −5.87539765226068021129552459433, −5.01606015799933394506368707227, −4.62057462413219048144377901220, −3.14680610531792933908741063349, −2.25751955565002003201830725182, −1.18546687341616271534370964652,
1.18546687341616271534370964652, 2.25751955565002003201830725182, 3.14680610531792933908741063349, 4.62057462413219048144377901220, 5.01606015799933394506368707227, 5.87539765226068021129552459433, 6.52821472562418635970661971261, 7.18289590960962294788035624583, 8.201977283420626733633363323149, 9.387240155422170765654729871172
