| L(s) = 1 | + 2·3-s − 5-s + 9-s − 4·13-s − 2·15-s − 4·19-s − 6·23-s + 25-s − 4·27-s + 6·29-s − 8·31-s + 2·37-s − 8·39-s + 6·41-s − 8·43-s − 45-s − 6·47-s − 6·53-s − 8·57-s + 12·59-s + 2·61-s + 4·65-s − 10·67-s − 12·69-s − 12·71-s − 16·73-s + 2·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.10·13-s − 0.516·15-s − 0.917·19-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.875·47-s − 0.824·53-s − 1.05·57-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 1.22·67-s − 1.44·69-s − 1.42·71-s − 1.87·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7112393220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7112393220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.64960565888392, −13.02800091442533, −12.79915344953149, −12.08200795417330, −11.78219938782164, −11.17451149613784, −10.60926426949664, −9.934105639361232, −9.769036843025191, −9.064653767303942, −8.575502507511158, −8.189094840904036, −7.810235430049475, −7.111006731912160, −6.892350078263692, −5.921628272077579, −5.636634662863204, −4.634338336761822, −4.384884105898293, −3.758467460147803, −3.082488189609871, −2.686450393388315, −2.024262804546562, −1.505061368567583, −0.2201720538458633,
0.2201720538458633, 1.505061368567583, 2.024262804546562, 2.686450393388315, 3.082488189609871, 3.758467460147803, 4.384884105898293, 4.634338336761822, 5.636634662863204, 5.921628272077579, 6.892350078263692, 7.111006731912160, 7.810235430049475, 8.189094840904036, 8.575502507511158, 9.064653767303942, 9.769036843025191, 9.934105639361232, 10.60926426949664, 11.17451149613784, 11.78219938782164, 12.08200795417330, 12.79915344953149, 13.02800091442533, 13.64960565888392
