| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 15-s + 2·17-s + 8·19-s − 4·21-s + 4·23-s + 25-s + 27-s − 6·29-s + 8·31-s − 4·35-s + 6·37-s + 6·41-s + 4·43-s + 45-s − 4·47-s + 9·49-s + 2·51-s − 6·53-s + 8·57-s + 8·59-s + 10·61-s − 4·63-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.258·15-s + 0.485·17-s + 1.83·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s + 1.05·57-s + 1.04·59-s + 1.28·61-s − 0.503·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.115649448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.115649448\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.24495028328749, −12.91510266465019, −12.49092948787037, −11.87001722694607, −11.44580750760063, −10.79466898177860, −10.25243000084183, −9.680579677621331, −9.453005731869870, −9.286778264741548, −8.483633828026592, −7.820712846984448, −7.556194555964191, −6.776665812606671, −6.592386404865346, −5.857016954054467, −5.440933773550397, −4.887166165847951, −4.062285566575007, −3.578723244296199, −2.988297680709934, −2.753233649260903, −1.991815799423307, −1.025088107988275, −0.6769186859013915,
0.6769186859013915, 1.025088107988275, 1.991815799423307, 2.753233649260903, 2.988297680709934, 3.578723244296199, 4.062285566575007, 4.887166165847951, 5.440933773550397, 5.857016954054467, 6.592386404865346, 6.776665812606671, 7.556194555964191, 7.820712846984448, 8.483633828026592, 9.286778264741548, 9.453005731869870, 9.680579677621331, 10.25243000084183, 10.79466898177860, 11.44580750760063, 11.87001722694607, 12.49092948787037, 12.91510266465019, 13.24495028328749
