| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 4·13-s − 15-s − 4·19-s + 8·23-s + 25-s + 27-s − 4·29-s + 4·33-s + 4·37-s + 4·39-s + 10·41-s + 4·43-s − 45-s − 8·47-s − 7·49-s + 4·53-s − 4·55-s − 4·57-s + 12·59-s + 4·61-s − 4·65-s + 12·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.258·15-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 0.696·33-s + 0.657·37-s + 0.640·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s − 49-s + 0.549·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 0.512·61-s − 0.496·65-s + 1.46·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.855226751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.855226751\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.00061045863885, −12.66570681406562, −11.96324461271636, −11.34660864371766, −11.19005884522279, −10.76313649683607, −10.09772632868903, −9.398139599350161, −9.220651311103007, −8.730636743178413, −8.240663421353350, −7.871473446904787, −7.185073367438893, −6.712349770324797, −6.419422448243589, −5.763524885468620, −5.163936203311423, −4.421978450355222, −4.122684298141909, −3.525663169300589, −3.214041282537525, −2.371184663054479, −1.861262149014054, −1.033464447801716, −0.7004348438910588,
0.7004348438910588, 1.033464447801716, 1.861262149014054, 2.371184663054479, 3.214041282537525, 3.525663169300589, 4.122684298141909, 4.421978450355222, 5.163936203311423, 5.763524885468620, 6.419422448243589, 6.712349770324797, 7.185073367438893, 7.871473446904787, 8.240663421353350, 8.730636743178413, 9.220651311103007, 9.398139599350161, 10.09772632868903, 10.76313649683607, 11.19005884522279, 11.34660864371766, 11.96324461271636, 12.66570681406562, 13.00061045863885
