| L(s) = 1 | − 4·7-s − 3·9-s − 2·13-s + 2·17-s − 4·19-s − 4·23-s + 2·29-s − 8·31-s − 6·37-s + 6·41-s − 8·43-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s + 2·61-s + 12·63-s − 8·67-s − 6·73-s + 9·81-s − 16·83-s − 6·89-s + 8·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.977·67-s − 0.702·73-s + 81-s − 1.75·83-s − 0.635·89-s + 0.838·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.11384881009220, −15.43914869899232, −14.79863328628902, −14.35555022787168, −13.87490401254883, −13.09282142133417, −12.80103561443314, −12.19209953465276, −11.74742941511285, −11.02636690003311, −10.36558441197602, −10.00702047379965, −9.273611048599304, −8.945118890370354, −8.187462407886080, −7.632254256433091, −6.811530387540573, −6.429973736623183, −5.762032815613902, −5.306321920457651, −4.362313728852970, −3.637744193130045, −3.076688425789851, −2.490866461653527, −1.569184152334495, 0, 0,
1.569184152334495, 2.490866461653527, 3.076688425789851, 3.637744193130045, 4.362313728852970, 5.306321920457651, 5.762032815613902, 6.429973736623183, 6.811530387540573, 7.632254256433091, 8.187462407886080, 8.945118890370354, 9.273611048599304, 10.00702047379965, 10.36558441197602, 11.02636690003311, 11.74742941511285, 12.19209953465276, 12.80103561443314, 13.09282142133417, 13.87490401254883, 14.35555022787168, 14.79863328628902, 15.43914869899232, 16.11384881009220
