| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·12-s + 16-s + 6·17-s + 18-s − 6·19-s − 23-s − 2·24-s − 5·25-s + 4·27-s − 10·29-s − 4·31-s + 32-s + 6·34-s + 36-s − 2·37-s − 6·38-s − 10·41-s + 4·43-s − 46-s − 12·47-s − 2·48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.208·23-s − 0.408·24-s − 25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.973·38-s − 1.56·41-s + 0.609·43-s − 0.147·46-s − 1.75·47-s − 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.12845693086714, −12.69798506899573, −12.33151104983003, −11.78982432051386, −11.51982011162680, −11.09544230955230, −10.55109604855267, −10.23826872907971, −9.692142981040777, −9.178223400740502, −8.449626779347890, −8.039426890854924, −7.462548807188925, −7.046018586694465, −6.349217559482741, −6.129797902945494, −5.490679877731573, −5.353935454629644, −4.688590350288101, −4.167050343122565, −3.508947723675445, −3.262523558224759, −2.295763197530794, −1.759469748026957, −1.234518943245509, 0, 0,
1.234518943245509, 1.759469748026957, 2.295763197530794, 3.262523558224759, 3.508947723675445, 4.167050343122565, 4.688590350288101, 5.353935454629644, 5.490679877731573, 6.129797902945494, 6.349217559482741, 7.046018586694465, 7.462548807188925, 8.039426890854924, 8.449626779347890, 9.178223400740502, 9.692142981040777, 10.23826872907971, 10.55109604855267, 11.09544230955230, 11.51982011162680, 11.78982432051386, 12.33151104983003, 12.69798506899573, 13.12845693086714
