L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 6·13-s − 2·15-s − 2·17-s + 4·19-s + 21-s − 25-s + 27-s + 2·29-s − 8·31-s − 2·35-s + 6·37-s − 6·39-s − 10·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s − 2·51-s + 6·53-s + 4·57-s − 4·59-s + 10·61-s + 63-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s − 0.960·39-s − 1.56·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s + 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.97604097102439, −14.67217958681097, −14.01191538875626, −13.57889854865678, −12.91945027799115, −12.37286126958121, −11.85293252241708, −11.56008466980058, −10.86258553333696, −10.23081221583596, −9.703905894679903, −9.197804657807305, −8.630013699579708, −7.886945374438881, −7.692335698581459, −7.079694817561442, −6.635834844392715, −5.560172240438497, −5.056374565047149, −4.539757157321770, −3.784202602546271, −3.371644950630349, −2.423070679705750, −2.054356506242279, −0.9166070569173201, 0,
0.9166070569173201, 2.054356506242279, 2.423070679705750, 3.371644950630349, 3.784202602546271, 4.539757157321770, 5.056374565047149, 5.560172240438497, 6.635834844392715, 7.079694817561442, 7.692335698581459, 7.886945374438881, 8.630013699579708, 9.197804657807305, 9.703905894679903, 10.23081221583596, 10.86258553333696, 11.56008466980058, 11.85293252241708, 12.37286126958121, 12.91945027799115, 13.57889854865678, 14.01191538875626, 14.67217958681097, 14.97604097102439