L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 9-s − 11-s − 4·15-s + 4·17-s + 4·19-s − 2·21-s + 4·23-s − 25-s − 4·27-s − 2·29-s + 2·31-s − 2·33-s + 2·35-s + 6·37-s + 4·41-s − 4·43-s − 2·45-s − 2·47-s + 49-s + 8·51-s − 2·53-s + 2·55-s + 8·57-s − 6·59-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.03·15-s + 0.970·17-s + 0.917·19-s − 0.436·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 0.371·29-s + 0.359·31-s − 0.348·33-s + 0.338·35-s + 0.986·37-s + 0.624·41-s − 0.609·43-s − 0.298·45-s − 0.291·47-s + 1/7·49-s + 1.12·51-s − 0.274·53-s + 0.269·55-s + 1.05·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.247635574\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.247635574\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.036991817391048098082022022747, −7.79176991605327917560108987491, −7.13469277269583351110067947880, −6.11939333308679473761121471279, −5.26631240023283908554477437321, −4.36235253180583064087154778916, −3.34630364038171409907826355088, −3.21848782855454117736763082704, −2.11705595583072934562648531412, −0.76910974709403191668055663643,
0.76910974709403191668055663643, 2.11705595583072934562648531412, 3.21848782855454117736763082704, 3.34630364038171409907826355088, 4.36235253180583064087154778916, 5.26631240023283908554477437321, 6.11939333308679473761121471279, 7.13469277269583351110067947880, 7.79176991605327917560108987491, 8.036991817391048098082022022747