L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 7-s + 2·10-s − 4·13-s + 2·14-s − 4·16-s − 17-s + 2·20-s − 4·23-s − 4·25-s − 8·26-s + 2·28-s − 2·31-s − 8·32-s − 2·34-s + 35-s + 6·37-s + 2·41-s − 3·43-s − 8·46-s − 7·47-s + 49-s − 8·50-s − 8·52-s − 12·53-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.377·7-s + 0.632·10-s − 1.10·13-s + 0.534·14-s − 16-s − 0.242·17-s + 0.447·20-s − 0.834·23-s − 4/5·25-s − 1.56·26-s + 0.377·28-s − 0.359·31-s − 1.41·32-s − 0.342·34-s + 0.169·35-s + 0.986·37-s + 0.312·41-s − 0.457·43-s − 1.17·46-s − 1.02·47-s + 1/7·49-s − 1.13·50-s − 1.10·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−7.40080828547802094485935924406, −6.52463250700936829356519762561, −5.99561438017873514773677102190, −5.31249529338559192633761326421, −4.68871770003722914955265480902, −4.12959384390629203071796864161, −3.22561675761706290269562957932, −2.43015391780097358738625301816, −1.73485302021313472442264248997, 0,
1.73485302021313472442264248997, 2.43015391780097358738625301816, 3.22561675761706290269562957932, 4.12959384390629203071796864161, 4.68871770003722914955265480902, 5.31249529338559192633761326421, 5.99561438017873514773677102190, 6.52463250700936829356519762561, 7.40080828547802094485935924406