| L(s) = 1 | + 3-s + 9-s + 2·11-s + 4·13-s + 2·17-s + 2·19-s + 4·23-s + 27-s + 6·29-s − 2·31-s + 2·33-s − 10·37-s + 4·39-s + 10·41-s + 12·43-s + 8·47-s + 2·51-s + 2·57-s − 8·59-s + 2·61-s − 12·67-s + 4·69-s + 10·71-s + 4·73-s + 81-s + 12·83-s + 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.485·17-s + 0.458·19-s + 0.834·23-s + 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.348·33-s − 1.64·37-s + 0.640·39-s + 1.56·41-s + 1.82·43-s + 1.16·47-s + 0.280·51-s + 0.264·57-s − 1.04·59-s + 0.256·61-s − 1.46·67-s + 0.481·69-s + 1.18·71-s + 0.468·73-s + 1/9·81-s + 1.31·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.529209259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.529209259\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.17556849526344, −13.97169590771729, −13.44873560846072, −12.84742411022125, −12.17171388667569, −12.08902318938495, −11.11119579077560, −10.76437054007407, −10.37634611808504, −9.437890880649125, −9.204405503970207, −8.766290782095648, −8.093018238007638, −7.634451539681518, −7.000171293532126, −6.528154950904671, −5.811329992354949, −5.386973842249589, −4.487136273289316, −4.043891985267183, −3.373998427314678, −2.905170390601622, −2.105662679220935, −1.249857694129346, −0.8036859856081205,
0.8036859856081205, 1.249857694129346, 2.105662679220935, 2.905170390601622, 3.373998427314678, 4.043891985267183, 4.487136273289316, 5.386973842249589, 5.811329992354949, 6.528154950904671, 7.000171293532126, 7.634451539681518, 8.093018238007638, 8.766290782095648, 9.204405503970207, 9.437890880649125, 10.37634611808504, 10.76437054007407, 11.11119579077560, 12.08902318938495, 12.17171388667569, 12.84742411022125, 13.44873560846072, 13.97169590771729, 14.17556849526344
