L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 3.08·7-s − 8-s + 9-s − 10-s + 11-s + 12-s − 6.59·13-s + 3.08·14-s + 15-s + 16-s − 17-s − 18-s + 1.57·19-s + 20-s − 3.08·21-s − 22-s + 0.916·23-s − 24-s + 25-s + 6.59·26-s + 27-s − 3.08·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.16·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.82·13-s + 0.824·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.361·19-s + 0.223·20-s − 0.672·21-s − 0.213·22-s + 0.191·23-s − 0.204·24-s + 0.200·25-s + 1.29·26-s + 0.192·27-s − 0.582·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.324674637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.324674637\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 3.08T + 7T^{2} \) |
| 13 | \( 1 + 6.59T + 13T^{2} \) |
| 19 | \( 1 - 1.57T + 19T^{2} \) |
| 23 | \( 1 - 0.916T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 3.57T + 73T^{2} \) |
| 79 | \( 1 - 2.42T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 4.16T + 89T^{2} \) |
| 97 | \( 1 + 2.65T + 97T^{2} \) |
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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
−8.147163505248183710055062381681, −7.46007702530101620490034254376, −6.82125357399250336799312645375, −6.29955477069300677514511140333, −5.30901625008912747662616238868, −4.45740381229890052276477521879, −3.34595464740307770737935931152, −2.70031552557678606815138141538, −1.99084372754945234977751152957, −0.63689071700793022988470663843,
0.63689071700793022988470663843, 1.99084372754945234977751152957, 2.70031552557678606815138141538, 3.34595464740307770737935931152, 4.45740381229890052276477521879, 5.30901625008912747662616238868, 6.29955477069300677514511140333, 6.82125357399250336799312645375, 7.46007702530101620490034254376, 8.147163505248183710055062381681