L(s) = 1 | − 2-s − 3-s + 4-s − 2.37·5-s + 6-s + 0.603·7-s − 8-s + 9-s + 2.37·10-s − 1.86·11-s − 12-s − 6.32·13-s − 0.603·14-s + 2.37·15-s + 16-s + 4.04·17-s − 18-s − 19-s − 2.37·20-s − 0.603·21-s + 1.86·22-s − 5.46·23-s + 24-s + 0.649·25-s + 6.32·26-s − 27-s + 0.603·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.06·5-s + 0.408·6-s + 0.227·7-s − 0.353·8-s + 0.333·9-s + 0.751·10-s − 0.562·11-s − 0.288·12-s − 1.75·13-s − 0.161·14-s + 0.613·15-s + 0.250·16-s + 0.980·17-s − 0.235·18-s − 0.229·19-s − 0.531·20-s − 0.131·21-s + 0.397·22-s − 1.14·23-s + 0.204·24-s + 0.129·25-s + 1.24·26-s − 0.192·27-s + 0.113·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2876632902\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2876632902\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 0.603T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 - 4.04T + 17T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 9.07T + 29T^{2} \) |
| 31 | \( 1 + 3.31T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 + 8.17T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.22T + 73T^{2} \) |
| 79 | \( 1 + 8.75T + 79T^{2} \) |
| 83 | \( 1 + 7.83T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 + 9.54T + 97T^{2} \) |
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
−8.149026359028187689943422800174, −7.34995443915296430052459009773, −7.01181883936245115397842657292, −5.99751694941220840345072616747, −5.14434340936114418923875281157, −4.59465667557258504473010844126, −3.59484338240507972579166440299, −2.69743345209173527885264808550, −1.66530426499429613251523338973, −0.31599889151716148770017745970,
0.31599889151716148770017745970, 1.66530426499429613251523338973, 2.69743345209173527885264808550, 3.59484338240507972579166440299, 4.59465667557258504473010844126, 5.14434340936114418923875281157, 5.99751694941220840345072616747, 7.01181883936245115397842657292, 7.34995443915296430052459009773, 8.149026359028187689943422800174