L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 3·11-s − 13-s + 15-s + 3·17-s + 6·19-s + 21-s + 23-s + 25-s + 27-s + 4·29-s + 8·31-s − 3·33-s + 35-s − 3·37-s − 39-s − 3·41-s + 6·43-s + 45-s − 12·47-s − 6·49-s + 3·51-s − 53-s − 3·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s + 0.727·17-s + 1.37·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.43·31-s − 0.522·33-s + 0.169·35-s − 0.493·37-s − 0.160·39-s − 0.468·41-s + 0.914·43-s + 0.149·45-s − 1.75·47-s − 6/7·49-s + 0.420·51-s − 0.137·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.996948797\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.996948797\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.047302069950730281574356946876, −7.49265990556339247265918395954, −6.74053198957974933118875394690, −5.84530554752435688182227970096, −5.08370319766216404367091124283, −4.61248178209170083285750463390, −3.33282812219536295115237024910, −2.87376314889479207438380090675, −1.90254529231889943391348065422, −0.908515222001293130349717551836,
0.908515222001293130349717551836, 1.90254529231889943391348065422, 2.87376314889479207438380090675, 3.33282812219536295115237024910, 4.61248178209170083285750463390, 5.08370319766216404367091124283, 5.84530554752435688182227970096, 6.74053198957974933118875394690, 7.49265990556339247265918395954, 8.047302069950730281574356946876