| L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·11-s − 16-s + 4·22-s − 8·23-s − 5·25-s − 2·29-s − 5·32-s − 6·37-s − 12·43-s + 4·44-s + 8·46-s + 5·50-s + 10·53-s + 2·58-s + 7·64-s + 4·67-s − 16·71-s + 6·74-s + 8·79-s + 12·86-s − 12·88-s + 8·92-s + 5·100-s − 10·106-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 1/4·16-s + 0.852·22-s − 1.66·23-s − 25-s − 0.371·29-s − 0.883·32-s − 0.986·37-s − 1.82·43-s + 0.603·44-s + 1.17·46-s + 0.707·50-s + 1.37·53-s + 0.262·58-s + 7/8·64-s + 0.488·67-s − 1.89·71-s + 0.697·74-s + 0.900·79-s + 1.29·86-s − 1.27·88-s + 0.834·92-s + 1/2·100-s − 0.971·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−10.22557678234009295397637516324, −10.00216059812775063040598752212, −8.772288861366294277115033481020, −8.063480898156861953425973667312, −7.31504642298749821815076432316, −5.85934501838508108877594225367, −4.87914944321891573254016142709, −3.69498409096789705984132005254, −1.99068740018288862581437794517, 0,
1.99068740018288862581437794517, 3.69498409096789705984132005254, 4.87914944321891573254016142709, 5.85934501838508108877594225367, 7.31504642298749821815076432316, 8.063480898156861953425973667312, 8.772288861366294277115033481020, 10.00216059812775063040598752212, 10.22557678234009295397637516324
