L(s) = 1 | + 2-s + 3-s + 4-s + 1.83·5-s + 6-s − 4.13·7-s + 8-s + 9-s + 1.83·10-s − 4.28·11-s + 12-s + 0.501·13-s − 4.13·14-s + 1.83·15-s + 16-s − 17-s + 18-s + 4.55·19-s + 1.83·20-s − 4.13·21-s − 4.28·22-s − 6.84·23-s + 24-s − 1.64·25-s + 0.501·26-s + 27-s − 4.13·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.819·5-s + 0.408·6-s − 1.56·7-s + 0.353·8-s + 0.333·9-s + 0.579·10-s − 1.29·11-s + 0.288·12-s + 0.138·13-s − 1.10·14-s + 0.473·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.04·19-s + 0.409·20-s − 0.902·21-s − 0.913·22-s − 1.42·23-s + 0.204·24-s − 0.328·25-s + 0.0982·26-s + 0.192·27-s − 0.781·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 13 | \( 1 - 0.501T + 13T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 + 3.33T + 29T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 5.78T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 61 | \( 1 - 3.98T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 6.04T + 71T^{2} \) |
| 73 | \( 1 + 6.23T + 73T^{2} \) |
| 79 | \( 1 - 9.82T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.64460156576727924367282549886, −6.89938625441091405173759445679, −6.16414247688147490920724427557, −5.67485157313699253595252550209, −4.91035751408397628720432928755, −3.82563453354281331858838946155, −3.19931978158229223853339151015, −2.55628191986725704637445220500, −1.72812162739170178230731661288, 0,
1.72812162739170178230731661288, 2.55628191986725704637445220500, 3.19931978158229223853339151015, 3.82563453354281331858838946155, 4.91035751408397628720432928755, 5.67485157313699253595252550209, 6.16414247688147490920724427557, 6.89938625441091405173759445679, 7.64460156576727924367282549886