L(s) = 1 | − 2-s − 2.86·3-s + 4-s + 5-s + 2.86·6-s − 1.78·7-s − 8-s + 5.19·9-s − 10-s + 11-s − 2.86·12-s − 5.69·13-s + 1.78·14-s − 2.86·15-s + 16-s + 0.963·17-s − 5.19·18-s + 4.63·19-s + 20-s + 5.12·21-s − 22-s − 7.69·23-s + 2.86·24-s + 25-s + 5.69·26-s − 6.28·27-s − 1.78·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.65·3-s + 0.5·4-s + 0.447·5-s + 1.16·6-s − 0.676·7-s − 0.353·8-s + 1.73·9-s − 0.316·10-s + 0.301·11-s − 0.826·12-s − 1.58·13-s + 0.478·14-s − 0.739·15-s + 0.250·16-s + 0.233·17-s − 1.22·18-s + 1.06·19-s + 0.223·20-s + 1.11·21-s − 0.213·22-s − 1.60·23-s + 0.584·24-s + 0.200·25-s + 1.11·26-s − 1.20·27-s − 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 2.86T + 3T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 - 0.963T + 17T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 4.30T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 61 | \( 1 - 3.05T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 6.67T + 73T^{2} \) |
| 79 | \( 1 + 6.14T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 + 3.42T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−7.66329545464333117770565097766, −7.17242372551812700355423994899, −6.34843348904324092267047891254, −5.98590093083542838492242965937, −5.13713378386452016179908879800, −4.49087389034731714289884729078, −3.20854368031342544361032400673, −2.13838263193937889778502939675, −0.978337269785765959093172738511, 0,
0.978337269785765959093172738511, 2.13838263193937889778502939675, 3.20854368031342544361032400673, 4.49087389034731714289884729078, 5.13713378386452016179908879800, 5.98590093083542838492242965937, 6.34843348904324092267047891254, 7.17242372551812700355423994899, 7.66329545464333117770565097766