| L(s) = 1 | + 3·7-s − 3·11-s + 13-s − 3·17-s + 4·19-s + 6·23-s − 29-s + 3·31-s + 10·37-s + 8·41-s − 10·43-s − 47-s + 2·49-s + 53-s + 7·59-s + 7·61-s + 11·67-s + 6·73-s − 9·77-s + 83-s − 2·89-s + 3·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s + 1.25·23-s − 0.185·29-s + 0.538·31-s + 1.64·37-s + 1.24·41-s − 1.52·43-s − 0.145·47-s + 2/7·49-s + 0.137·53-s + 0.911·59-s + 0.896·61-s + 1.34·67-s + 0.702·73-s − 1.02·77-s + 0.109·83-s − 0.211·89-s + 0.314·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.723432442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.723432442\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−15.22304920604986, −15.06719034977713, −14.41134634938790, −13.84748037416742, −13.15194674083873, −13.03189722965052, −12.12003426224628, −11.51172394690887, −11.05150264904922, −10.80899228646727, −9.856414445771073, −9.481362263725003, −8.686663694433545, −8.099691810898282, −7.828325036481202, −6.996713591895021, −6.522450515730793, −5.470006607719798, −5.244300690647106, −4.537770001419386, −3.904077058473122, −2.902480027888051, −2.416820149642887, −1.458443131920598, −0.7007855970428651,
0.7007855970428651, 1.458443131920598, 2.416820149642887, 2.902480027888051, 3.904077058473122, 4.537770001419386, 5.244300690647106, 5.470006607719798, 6.522450515730793, 6.996713591895021, 7.828325036481202, 8.099691810898282, 8.686663694433545, 9.481362263725003, 9.856414445771073, 10.80899228646727, 11.05150264904922, 11.51172394690887, 12.12003426224628, 13.03189722965052, 13.15194674083873, 13.84748037416742, 14.41134634938790, 15.06719034977713, 15.22304920604986
