L(s) = 1 | − 2·7-s − 11-s − 4·13-s + 5·17-s + 19-s − 2·23-s + 8·29-s + 10·31-s + 6·37-s + 3·41-s − 4·43-s + 4·47-s − 3·49-s + 6·53-s − 8·59-s + 10·61-s + 67-s + 12·71-s − 3·73-s + 2·77-s + 6·79-s − 13·83-s + 9·89-s + 8·91-s + 14·97-s − 6·101-s − 4·103-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 0.301·11-s − 1.10·13-s + 1.21·17-s + 0.229·19-s − 0.417·23-s + 1.48·29-s + 1.79·31-s + 0.986·37-s + 0.468·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.824·53-s − 1.04·59-s + 1.28·61-s + 0.122·67-s + 1.42·71-s − 0.351·73-s + 0.227·77-s + 0.675·79-s − 1.42·83-s + 0.953·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.460240514\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.460240514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.549345032844968179219680675990, −8.351506683319880487851775121042, −7.77075401088767705375118610734, −6.86378396127984906651881332641, −6.12495822792309039345594276838, −5.20481348804422316781252563482, −4.36172478777594960027231636860, −3.18472537975480801189708163896, −2.47956256564960138993576183683, −0.819426402868476389338501487291,
0.819426402868476389338501487291, 2.47956256564960138993576183683, 3.18472537975480801189708163896, 4.36172478777594960027231636860, 5.20481348804422316781252563482, 6.12495822792309039345594276838, 6.86378396127984906651881332641, 7.77075401088767705375118610734, 8.351506683319880487851775121042, 9.549345032844968179219680675990