L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5·5-s + 6·6-s − 2·7-s − 8·8-s + 9·9-s − 10·10-s − 16·11-s − 12·12-s − 10·13-s + 4·14-s − 15·15-s + 16·16-s + 36·17-s − 18·18-s + 19·19-s + 20·20-s + 6·21-s + 32·22-s + 124·23-s + 24·24-s + 25·25-s + 20·26-s − 27·27-s − 8·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.107·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.438·11-s − 0.288·12-s − 0.213·13-s + 0.0763·14-s − 0.258·15-s + 1/4·16-s + 0.513·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.0623·21-s + 0.310·22-s + 1.12·23-s + 0.204·24-s + 1/5·25-s + 0.150·26-s − 0.192·27-s − 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.095418746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.095418746\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 23 | \( 1 - 124 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 74 T + p^{3} T^{2} \) |
| 37 | \( 1 - 94 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 276 T + p^{3} T^{2} \) |
| 47 | \( 1 - 540 T + p^{3} T^{2} \) |
| 53 | \( 1 - 146 T + p^{3} T^{2} \) |
| 59 | \( 1 - 606 T + p^{3} T^{2} \) |
| 61 | \( 1 - 450 T + p^{3} T^{2} \) |
| 67 | \( 1 - 180 T + p^{3} T^{2} \) |
| 71 | \( 1 + 456 T + p^{3} T^{2} \) |
| 73 | \( 1 - 14 T + p^{3} T^{2} \) |
| 79 | \( 1 - 550 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1442 T + p^{3} T^{2} \) |
| 89 | \( 1 - 212 T + p^{3} T^{2} \) |
| 97 | \( 1 + 830 T + p^{3} T^{2} \) |
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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
−10.26111510297759284980215544969, −9.581936345231192845311452632418, −8.691209191358807201530437972780, −7.58174899138162550433320136205, −6.84776271533872306601395312948, −5.77457633679565542337552473519, −5.00749245156858223555424617183, −3.41113118980625084260031843664, −2.05909248793603210295368714780, −0.72014837846382018424340689433,
0.72014837846382018424340689433, 2.05909248793603210295368714780, 3.41113118980625084260031843664, 5.00749245156858223555424617183, 5.77457633679565542337552473519, 6.84776271533872306601395312948, 7.58174899138162550433320136205, 8.691209191358807201530437972780, 9.581936345231192845311452632418, 10.26111510297759284980215544969