L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 4·11-s + 12-s + 13-s − 15-s + 16-s + 18-s + 4·19-s − 20-s − 4·22-s − 8·23-s + 24-s + 25-s + 26-s + 27-s − 6·29-s − 30-s + 8·31-s + 32-s − 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.802624064\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.802624064\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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
−13.66580268423387, −13.08758085284170, −12.88593434298243, −12.24320418236043, −11.68979320500866, −11.37810534901730, −10.73967070691822, −10.24309152525413, −9.754964503684960, −9.292855170591476, −8.545223670798658, −7.961703101993921, −7.650767152811271, −7.432214452212716, −6.435363320482919, −6.033861336021173, −5.558337589118837, −4.777877163583452, −4.410308695211171, −3.839932485394473, −3.173050344069318, −2.756756057971685, −2.163020592098812, −1.409250148058395, −0.5006859080462561,
0.5006859080462561, 1.409250148058395, 2.163020592098812, 2.756756057971685, 3.173050344069318, 3.839932485394473, 4.410308695211171, 4.777877163583452, 5.558337589118837, 6.033861336021173, 6.435363320482919, 7.432214452212716, 7.650767152811271, 7.961703101993921, 8.545223670798658, 9.292855170591476, 9.754964503684960, 10.24309152525413, 10.73967070691822, 11.37810534901730, 11.68979320500866, 12.24320418236043, 12.88593434298243, 13.08758085284170, 13.66580268423387