L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 6·17-s − 18-s + 20-s + 21-s + 22-s + 8·23-s + 24-s − 4·25-s − 26-s − 27-s − 28-s − 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s + 0.213·22-s + 1.66·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−12.90412683632889, −12.44414347198568, −11.75099377738632, −11.52596709417684, −10.82806659151721, −10.67252546263016, −10.29548755296837, −9.587932022657605, −9.181689459152137, −8.879081538451502, −8.490963061134799, −7.611585413569046, −7.284975024103202, −6.916123027853398, −6.348839021543208, −5.911152326564077, −5.456600541303411, −4.933007140555527, −4.338140261738370, −3.701634119832523, −3.141504845222215, −2.480196472049342, −1.920222504909215, −1.441523437820403, −0.5935044739929845, 0,
0.5935044739929845, 1.441523437820403, 1.920222504909215, 2.480196472049342, 3.141504845222215, 3.701634119832523, 4.338140261738370, 4.933007140555527, 5.456600541303411, 5.911152326564077, 6.348839021543208, 6.916123027853398, 7.284975024103202, 7.611585413569046, 8.490963061134799, 8.879081538451502, 9.181689459152137, 9.587932022657605, 10.29548755296837, 10.67252546263016, 10.82806659151721, 11.52596709417684, 11.75099377738632, 12.44414347198568, 12.90412683632889