L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 2·13-s − 2·17-s − 4·19-s + 21-s + 8·23-s + 27-s − 6·29-s − 8·31-s − 33-s − 2·37-s − 2·39-s + 2·41-s − 4·43-s + 8·47-s + 49-s − 2·51-s − 10·53-s − 4·57-s − 12·59-s + 2·61-s + 63-s + 4·67-s + 8·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.963·69-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
\(L(1)\) |
\(\approx\) |
\(0.7456521515\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7456521515\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.62335709498998, −12.24322285506693, −11.40994785511167, −11.03746089149233, −10.87783584617932, −10.24103884931862, −9.710092008893580, −9.214600285921232, −8.878749652305698, −8.516138437157267, −7.899687951041020, −7.363277529690508, −7.181149127490809, −6.618086569788780, −5.927759308634099, −5.493678265181483, −4.844464798246992, −4.601917810391340, −3.875034290090583, −3.497350969084350, −2.636299003934731, −2.548695674768766, −1.611125289869130, −1.397431116774041, −0.1989655507303734,
0.1989655507303734, 1.397431116774041, 1.611125289869130, 2.548695674768766, 2.636299003934731, 3.497350969084350, 3.875034290090583, 4.601917810391340, 4.844464798246992, 5.493678265181483, 5.927759308634099, 6.618086569788780, 7.181149127490809, 7.363277529690508, 7.899687951041020, 8.516138437157267, 8.878749652305698, 9.214600285921232, 9.710092008893580, 10.24103884931862, 10.87783584617932, 11.03746089149233, 11.40994785511167, 12.24322285506693, 12.62335709498998