L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 2·13-s − 6·17-s + 2·19-s + 21-s + 27-s − 8·29-s − 8·31-s − 33-s + 6·37-s − 2·39-s − 8·41-s − 8·43-s − 10·47-s + 49-s − 6·51-s + 2·53-s + 2·57-s + 8·59-s − 10·61-s + 63-s − 2·67-s − 10·73-s − 77-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.218·21-s + 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.174·33-s + 0.986·37-s − 0.320·39-s − 1.24·41-s − 1.21·43-s − 1.45·47-s + 1/7·49-s − 0.840·51-s + 0.274·53-s + 0.264·57-s + 1.04·59-s − 1.28·61-s + 0.125·63-s − 0.244·67-s − 1.17·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2388940775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2388940775\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.62745097778100, −11.94161819201824, −11.60400538365589, −11.15852871384783, −10.73009319220344, −10.22573128006878, −9.656357744527172, −9.351998816426083, −8.854799407320382, −8.429200032854577, −7.941227475806608, −7.424506794243316, −7.128227748387764, −6.575873568471250, −6.028704884580249, −5.357793791745721, −4.984231853267456, −4.528200586970405, −3.880452493745993, −3.472378191899708, −2.843588931079495, −2.261314200207264, −1.805438977601864, −1.319732835305234, −0.1120321295756546,
0.1120321295756546, 1.319732835305234, 1.805438977601864, 2.261314200207264, 2.843588931079495, 3.472378191899708, 3.880452493745993, 4.528200586970405, 4.984231853267456, 5.357793791745721, 6.028704884580249, 6.575873568471250, 7.128227748387764, 7.424506794243316, 7.941227475806608, 8.429200032854577, 8.854799407320382, 9.351998816426083, 9.656357744527172, 10.22573128006878, 10.73009319220344, 11.15852871384783, 11.60400538365589, 11.94161819201824, 12.62745097778100