L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 3.41·11-s + 12-s + 15-s + 16-s − 1.41·17-s + 18-s + 2.82·19-s + 20-s + 3.41·22-s − 0.828·23-s + 24-s + 25-s + 27-s − 0.242·29-s + 30-s − 9.07·31-s + 32-s + 3.41·33-s − 1.41·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.02·11-s + 0.288·12-s + 0.258·15-s + 0.250·16-s − 0.342·17-s + 0.235·18-s + 0.648·19-s + 0.223·20-s + 0.727·22-s − 0.172·23-s + 0.204·24-s + 0.200·25-s + 0.192·27-s − 0.0450·29-s + 0.182·30-s − 1.62·31-s + 0.176·32-s + 0.594·33-s − 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.711713368\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.711713368\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 - 5.07T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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
−9.303923709418370639046652869945, −8.942791915843372980560375015824, −7.72207594628575358280964342281, −7.04413946226338617410435929933, −6.16937868360192706873339135053, −5.36564864907548915440145693109, −4.28316387354577249675123896734, −3.55145120842642612083412340212, −2.47654379521734336469500069073, −1.42548947774720625587042092820,
1.42548947774720625587042092820, 2.47654379521734336469500069073, 3.55145120842642612083412340212, 4.28316387354577249675123896734, 5.36564864907548915440145693109, 6.16937868360192706873339135053, 7.04413946226338617410435929933, 7.72207594628575358280964342281, 8.942791915843372980560375015824, 9.303923709418370639046652869945