| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s − 6·11-s + 12-s + 13-s − 4·14-s + 16-s − 4·17-s + 18-s + 2·19-s − 4·21-s − 6·22-s − 6·23-s + 24-s + 26-s + 27-s − 4·28-s − 10·29-s + 4·31-s + 32-s − 6·33-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 − 1.80·11-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 − 1.27·22-s − 1.25·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s − 1.04·33-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−8.872719737543440996534603832810, −7.76289295114252606314765360714, −7.32363828755953441216992328393, −6.22154667625865130162104138279, −5.70958952817226870085238408234, −4.59257511667402952390000494837, −3.65328685323855751887250626877, −2.90324884014687840380390019235, −2.13537777044490151588674952443, 0,
2.13537777044490151588674952443, 2.90324884014687840380390019235, 3.65328685323855751887250626877, 4.59257511667402952390000494837, 5.70958952817226870085238408234, 6.22154667625865130162104138279, 7.32363828755953441216992328393, 7.76289295114252606314765360714, 8.872719737543440996534603832810
