| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 6·11-s + 12-s − 15-s + 16-s − 17-s + 18-s + 3·19-s − 20-s − 6·22-s + 7·23-s + 24-s − 4·25-s + 27-s + 6·29-s − 30-s + 8·31-s + 32-s − 6·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.688·19-s − 0.223·20-s − 1.27·22-s + 1.45·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 1.11·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-s − 1.04·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.333254245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.333254245\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.188004153357926748152189322168, −7.50029631440256813271696032697, −6.93325692126481156908273617643, −5.97821071696668929818951852319, −5.08948017328861441743690271941, −4.66045481724531757267749386300, −3.65355222387770960690814634781, −2.84790132751660212354058306040, −2.37086097708569945522005508935, −0.866700547150567606676118007898,
0.866700547150567606676118007898, 2.37086097708569945522005508935, 2.84790132751660212354058306040, 3.65355222387770960690814634781, 4.66045481724531757267749386300, 5.08948017328861441743690271941, 5.97821071696668929818951852319, 6.93325692126481156908273617643, 7.50029631440256813271696032697, 8.188004153357926748152189322168
