| L(s) = 1 | + 7-s − 3·9-s + 2·11-s + 4·13-s − 2·17-s + 4·19-s + 4·23-s − 5·25-s + 6·29-s + 8·31-s − 2·37-s − 2·41-s + 10·43-s + 49-s + 2·53-s − 8·59-s + 8·61-s − 3·63-s + 2·67-s − 14·73-s + 2·77-s + 4·79-s + 9·81-s + 12·83-s − 6·89-s + 4·91-s + 6·97-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 0.603·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s + 0.834·23-s − 25-s + 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 1.52·43-s + 1/7·49-s + 0.274·53-s − 1.04·59-s + 1.02·61-s − 0.377·63-s + 0.244·67-s − 1.63·73-s + 0.227·77-s + 0.450·79-s + 81-s + 1.31·83-s − 0.635·89-s + 0.419·91-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666125043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666125043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 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
−10.14915575528512744704424398506, −9.085159879837952129396462991784, −8.556307069725046658734814323649, −7.69109110649490996887606564749, −6.54777995651802937371843140226, −5.85219222106469301830112911893, −4.81676464111172637776059220224, −3.71735389440188374007732407704, −2.65075041611570461759691498837, −1.10812390596012155410580072135,
1.10812390596012155410580072135, 2.65075041611570461759691498837, 3.71735389440188374007732407704, 4.81676464111172637776059220224, 5.85219222106469301830112911893, 6.54777995651802937371843140226, 7.69109110649490996887606564749, 8.556307069725046658734814323649, 9.085159879837952129396462991784, 10.14915575528512744704424398506
