| L(s) = 1 | − 2-s + 4-s + 3·5-s − 7-s − 8-s − 3·10-s − 3·11-s − 4·13-s + 14-s + 16-s + 2·19-s + 3·20-s + 3·22-s − 6·23-s + 4·25-s + 4·26-s − 28-s + 6·29-s + 5·31-s − 32-s − 3·35-s + 2·37-s − 2·38-s − 3·40-s − 6·41-s − 10·43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s − 0.904·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.458·19-s + 0.670·20-s + 0.639·22-s − 1.25·23-s + 4/5·25-s + 0.784·26-s − 0.188·28-s + 1.11·29-s + 0.898·31-s − 0.176·32-s − 0.507·35-s + 0.328·37-s − 0.324·38-s − 0.474·40-s − 0.937·41-s − 1.52·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7015748253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7015748253\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−15.58696880941000439507698489650, −14.21020099102705153396393794850, −13.18302012248397519939912394649, −11.91184860007480462763991644584, −10.11526477822370291644410044895, −9.872045258839871807563899217385, −8.268048909391662018895990050852, −6.73860900674537002760989556093, −5.37363015537505503550836847886, −2.44850978333263328730546301919,
2.44850978333263328730546301919, 5.37363015537505503550836847886, 6.73860900674537002760989556093, 8.268048909391662018895990050852, 9.872045258839871807563899217385, 10.11526477822370291644410044895, 11.91184860007480462763991644584, 13.18302012248397519939912394649, 14.21020099102705153396393794850, 15.58696880941000439507698489650
