L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 6·13-s − 6·17-s + 2·19-s + 21-s + 6·23-s − 27-s − 2·29-s − 6·31-s + 33-s + 4·37-s + 6·39-s − 4·41-s + 12·43-s + 12·47-s + 49-s + 6·51-s + 10·53-s − 2·57-s + 4·59-s + 2·61-s − 63-s + 4·67-s − 6·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 1.45·17-s + 0.458·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s − 0.371·29-s − 1.07·31-s + 0.174·33-s + 0.657·37-s + 0.960·39-s − 0.624·41-s + 1.82·43-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 1.37·53-s − 0.264·57-s + 0.520·59-s + 0.256·61-s − 0.125·63-s + 0.488·67-s − 0.722·69-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
\(L(1)\) |
\(\approx\) |
\(1.653944589\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.653944589\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.52618664727285, −12.10994550937221, −11.51128927083012, −11.19025285785883, −10.73612200197250, −10.29233907360886, −9.800094229861112, −9.337466860704941, −8.952577148022736, −8.567639163095051, −7.624563616463581, −7.383744393898126, −7.034956095203057, −6.605915702982348, −5.871322689012150, −5.485244764600889, −5.086682809239897, −4.479214906304524, −4.146691109549467, −3.454885404760819, −2.702155397000907, −2.382628794036468, −1.848032119181547, −0.7742100158454484, −0.4662426293048429,
0.4662426293048429, 0.7742100158454484, 1.848032119181547, 2.382628794036468, 2.702155397000907, 3.454885404760819, 4.146691109549467, 4.479214906304524, 5.086682809239897, 5.485244764600889, 5.871322689012150, 6.605915702982348, 7.034956095203057, 7.383744393898126, 7.624563616463581, 8.567639163095051, 8.952577148022736, 9.337466860704941, 9.800094229861112, 10.29233907360886, 10.73612200197250, 11.19025285785883, 11.51128927083012, 12.10994550937221, 12.52618664727285