| L(s) = 1 | + 3-s + 2·7-s + 9-s + 4·11-s + 2·17-s + 6·19-s + 2·21-s + 27-s + 10·29-s + 2·31-s + 4·33-s − 8·37-s + 2·41-s − 4·43-s + 8·47-s − 3·49-s + 2·51-s − 2·53-s + 6·57-s + 10·61-s + 2·63-s + 2·67-s + 12·71-s + 8·73-s + 8·77-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.485·17-s + 1.37·19-s + 0.436·21-s + 0.192·27-s + 1.85·29-s + 0.359·31-s + 0.696·33-s − 1.31·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.280·51-s − 0.274·53-s + 0.794·57-s + 1.28·61-s + 0.251·63-s + 0.244·67-s + 1.42·71-s + 0.936·73-s + 0.911·77-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.511555197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.511555197\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.80652030865101, −13.58636842355495, −12.55982079086195, −12.30980544533753, −11.81260243871510, −11.39058079719907, −10.83680662811264, −10.14952089406782, −9.785338186066571, −9.291295650919280, −8.699377029780463, −8.270850894751878, −7.885643402654851, −7.149944787459951, −6.803875385719582, −6.235683544379249, −5.412568772215351, −5.037751222777814, −4.403746694610616, −3.785517843697916, −3.304295266814597, −2.662559336282300, −1.913258560411195, −1.232071454936600, −0.8074078994010238,
0.8074078994010238, 1.232071454936600, 1.913258560411195, 2.662559336282300, 3.304295266814597, 3.785517843697916, 4.403746694610616, 5.037751222777814, 5.412568772215351, 6.235683544379249, 6.803875385719582, 7.149944787459951, 7.885643402654851, 8.270850894751878, 8.699377029780463, 9.291295650919280, 9.785338186066571, 10.14952089406782, 10.83680662811264, 11.39058079719907, 11.81260243871510, 12.30980544533753, 12.55982079086195, 13.58636842355495, 13.80652030865101
