| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 2·11-s − 12-s + 13-s + 4·14-s + 16-s + 4·17-s + 18-s + 2·19-s − 4·21-s + 2·22-s − 6·23-s − 24-s + 26-s − 27-s + 4·28-s − 2·29-s − 4·31-s + 32-s − 2·33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.426·22-s − 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 0.348·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.949359271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.949359271\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.182648124608353586531524291120, −8.182390005868430302670141391865, −7.56623418983499887240476159616, −6.74115278039220983750967218841, −5.63860941055669312891125420954, −5.33749202386514005593291265832, −4.28561933094907687769586580986, −3.63655378791011270375144157582, −2.10777338399542844214176104623, −1.20182002010796803067299954323,
1.20182002010796803067299954323, 2.10777338399542844214176104623, 3.63655378791011270375144157582, 4.28561933094907687769586580986, 5.33749202386514005593291265832, 5.63860941055669312891125420954, 6.74115278039220983750967218841, 7.56623418983499887240476159616, 8.182390005868430302670141391865, 9.182648124608353586531524291120
