L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 4·11-s + 2·13-s − 4·15-s + 2·17-s − 2·19-s − 2·23-s − 25-s + 4·27-s − 2·29-s + 2·31-s + 8·33-s − 10·37-s − 4·39-s − 2·41-s − 4·43-s + 2·45-s + 12·47-s − 7·49-s − 4·51-s − 8·55-s + 4·57-s − 12·59-s + 10·61-s + 4·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s − 0.458·19-s − 0.417·23-s − 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.359·31-s + 1.39·33-s − 1.64·37-s − 0.640·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.75·47-s − 49-s − 0.560·51-s − 1.07·55-s + 0.529·57-s − 1.56·59-s + 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9714276134\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9714276134\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 53 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.16334263816642, −12.65762516985334, −12.14652636094171, −11.85431206660995, −11.15776290440204, −10.73872576853538, −10.47895625929035, −9.943648974741281, −9.602010245369537, −8.789293760960893, −8.438635618561909, −7.891884147767405, −7.229818445906361, −6.787425418770882, −6.012901065060300, −5.950540941993550, −5.427168025707606, −4.916280553655297, −4.502158542646264, −3.550124447910175, −3.171102939332251, −2.233801364931888, −1.929656012055265, −1.069941010715623, −0.3301097551772842,
0.3301097551772842, 1.069941010715623, 1.929656012055265, 2.233801364931888, 3.171102939332251, 3.550124447910175, 4.502158542646264, 4.916280553655297, 5.427168025707606, 5.950540941993550, 6.012901065060300, 6.787425418770882, 7.229818445906361, 7.891884147767405, 8.438635618561909, 8.789293760960893, 9.602010245369537, 9.943648974741281, 10.47895625929035, 10.73872576853538, 11.15776290440204, 11.85431206660995, 12.14652636094171, 12.65762516985334, 13.16334263816642