L(s) = 1 | + 3·7-s − 3·11-s − 13-s − 3·17-s + 4·19-s − 3·23-s + 10·29-s − 6·31-s + 5·37-s − 5·41-s − 2·43-s + 2·47-s + 2·49-s − 11·53-s − 4·59-s + 61-s + 4·67-s − 3·71-s − 6·73-s − 9·77-s + 3·79-s + 16·83-s − 7·89-s − 3·91-s + 19·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 1.13·7-s − 0.904·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s − 0.625·23-s + 1.85·29-s − 1.07·31-s + 0.821·37-s − 0.780·41-s − 0.304·43-s + 0.291·47-s + 2/7·49-s − 1.51·53-s − 0.520·59-s + 0.128·61-s + 0.488·67-s − 0.356·71-s − 0.702·73-s − 1.02·77-s + 0.337·79-s + 1.75·83-s − 0.741·89-s − 0.314·91-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−15.70230195576251, −15.26210818187599, −14.51838089088018, −14.19578023284247, −13.63560144161540, −13.05601252081945, −12.48489718712257, −11.75607223244771, −11.51542645205856, −10.71336313978604, −10.41833484712242, −9.686139710240585, −9.071613084609743, −8.394594567037122, −7.871901820950275, −7.544511611327502, −6.693594895691091, −6.097757634283613, −5.207967989422648, −4.921460570967855, −4.304086047930634, −3.381504616632268, −2.630422938559312, −1.956808692006246, −1.136837913679120, 0,
1.136837913679120, 1.956808692006246, 2.630422938559312, 3.381504616632268, 4.304086047930634, 4.921460570967855, 5.207967989422648, 6.097757634283613, 6.693594895691091, 7.544511611327502, 7.871901820950275, 8.394594567037122, 9.071613084609743, 9.686139710240585, 10.41833484712242, 10.71336313978604, 11.51542645205856, 11.75607223244771, 12.48489718712257, 13.05601252081945, 13.63560144161540, 14.19578023284247, 14.51838089088018, 15.26210818187599, 15.70230195576251