| L(s) = 1 | + 0.0993·2-s + 3-s − 1.99·4-s + 1.12·5-s + 0.0993·6-s + 0.411·7-s − 0.396·8-s + 9-s + 0.112·10-s + 2.25·11-s − 1.99·12-s − 2.41·13-s + 0.0408·14-s + 1.12·15-s + 3.94·16-s − 17-s + 0.0993·18-s − 6.49·19-s − 2.24·20-s + 0.411·21-s + 0.223·22-s − 6.81·23-s − 0.396·24-s − 3.72·25-s − 0.239·26-s + 27-s − 0.818·28-s + ⋯ |
| L(s) = 1 | + 0.0702·2-s + 0.577·3-s − 0.995·4-s + 0.505·5-s + 0.0405·6-s + 0.155·7-s − 0.140·8-s + 0.333·9-s + 0.0355·10-s + 0.679·11-s − 0.574·12-s − 0.669·13-s + 0.0109·14-s + 0.291·15-s + 0.985·16-s − 0.242·17-s + 0.0234·18-s − 1.48·19-s − 0.502·20-s + 0.0897·21-s + 0.0477·22-s − 1.42·23-s − 0.0809·24-s − 0.744·25-s − 0.0470·26-s + 0.192·27-s − 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0993T + 2T^{2} \) |
| 5 | \( 1 - 1.12T + 5T^{2} \) |
| 7 | \( 1 - 0.411T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 19 | \( 1 + 6.49T + 19T^{2} \) |
| 23 | \( 1 + 6.81T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 + 1.60T + 37T^{2} \) |
| 41 | \( 1 - 4.87T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 7.65T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 83 | \( 1 - 2.81T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 9.24T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.220093504993379081859843406218, −7.55769133318606029798444723071, −6.43901266950975530898193920033, −5.96012301414795842104891799759, −4.80331316007922225838837543931, −4.33931610385529072874861244771, −3.54173576068043418986491536055, −2.42256219806459791620778172736, −1.55031875628636099806352946153, 0,
1.55031875628636099806352946153, 2.42256219806459791620778172736, 3.54173576068043418986491536055, 4.33931610385529072874861244771, 4.80331316007922225838837543931, 5.96012301414795842104891799759, 6.43901266950975530898193920033, 7.55769133318606029798444723071, 8.220093504993379081859843406218
