| L(s) = 1 | + 2·3-s + 7-s + 9-s − 5·11-s + 13-s + 4·17-s + 7·19-s + 2·21-s − 4·27-s + 5·29-s − 2·31-s − 10·33-s + 2·37-s + 2·39-s + 11·41-s + 43-s + 8·47-s − 6·49-s + 8·51-s + 14·57-s + 14·59-s − 10·61-s + 63-s − 8·67-s + 10·71-s + 7·73-s − 5·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s + 0.970·17-s + 1.60·19-s + 0.436·21-s − 0.769·27-s + 0.928·29-s − 0.359·31-s − 1.74·33-s + 0.328·37-s + 0.320·39-s + 1.71·41-s + 0.152·43-s + 1.16·47-s − 6/7·49-s + 1.12·51-s + 1.85·57-s + 1.82·59-s − 1.28·61-s + 0.125·63-s − 0.977·67-s + 1.18·71-s + 0.819·73-s − 0.569·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.866551165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.866551165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.00745842626360, −12.72330528525986, −12.10753974962872, −11.64044209198918, −11.04896208724363, −10.70255522775490, −10.03818389211253, −9.691006886434241, −9.295318758118139, −8.583238680411013, −8.319830526436510, −7.728158385698417, −7.522891486471332, −7.124097115579591, −6.086339680737491, −5.744325183831230, −5.210884622447702, −4.753778601371603, −4.021405350605338, −3.439717847452019, −2.956359360569365, −2.590438113327234, −2.003456823794605, −1.171779490024840, −0.6070515297217679,
0.6070515297217679, 1.171779490024840, 2.003456823794605, 2.590438113327234, 2.956359360569365, 3.439717847452019, 4.021405350605338, 4.753778601371603, 5.210884622447702, 5.744325183831230, 6.086339680737491, 7.124097115579591, 7.522891486471332, 7.728158385698417, 8.319830526436510, 8.583238680411013, 9.295318758118139, 9.691006886434241, 10.03818389211253, 10.70255522775490, 11.04896208724363, 11.64044209198918, 12.10753974962872, 12.72330528525986, 13.00745842626360
