| L(s) = 1 | + 4·2-s + 6·3-s + 16·4-s − 25·5-s + 24·6-s − 118·7-s + 64·8-s − 207·9-s − 100·10-s + 192·11-s + 96·12-s + 1.10e3·13-s − 472·14-s − 150·15-s + 256·16-s + 762·17-s − 828·18-s − 2.74e3·19-s − 400·20-s − 708·21-s + 768·22-s + 1.56e3·23-s + 384·24-s + 625·25-s + 4.42e3·26-s − 2.70e3·27-s − 1.88e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s − 0.447·5-s + 0.272·6-s − 0.910·7-s + 0.353·8-s − 0.851·9-s − 0.316·10-s + 0.478·11-s + 0.192·12-s + 1.81·13-s − 0.643·14-s − 0.172·15-s + 1/4·16-s + 0.639·17-s − 0.602·18-s − 1.74·19-s − 0.223·20-s − 0.350·21-s + 0.338·22-s + 0.617·23-s + 0.136·24-s + 1/5·25-s + 1.28·26-s − 0.712·27-s − 0.455·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.636778589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.636778589\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 118 T + p^{5} T^{2} \) |
| 11 | \( 1 - 192 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1106 T + p^{5} T^{2} \) |
| 17 | \( 1 - 762 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2740 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1566 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5910 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6868 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5518 T + p^{5} T^{2} \) |
| 41 | \( 1 + 378 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2434 T + p^{5} T^{2} \) |
| 47 | \( 1 - 13122 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9174 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34980 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9838 T + p^{5} T^{2} \) |
| 67 | \( 1 - 33722 T + p^{5} T^{2} \) |
| 71 | \( 1 - 70212 T + p^{5} T^{2} \) |
| 73 | \( 1 - 21986 T + p^{5} T^{2} \) |
| 79 | \( 1 - 4520 T + p^{5} T^{2} \) |
| 83 | \( 1 + 109074 T + p^{5} T^{2} \) |
| 89 | \( 1 - 38490 T + p^{5} T^{2} \) |
| 97 | \( 1 + 1918 T + p^{5} T^{2} \) |
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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
−20.02583321858768024715850251742, −18.96460454296572475775664511811, −16.78112915056809372972104844903, −15.47582110605087786463439853895, −14.06479657820184118254419041508, −12.67572670568411380283492600087, −10.99344907685009047379664891255, −8.643237327021262502631021576799, −6.30214796436421780881622992328, −3.52461673905853406661457795197,
3.52461673905853406661457795197, 6.30214796436421780881622992328, 8.643237327021262502631021576799, 10.99344907685009047379664891255, 12.67572670568411380283492600087, 14.06479657820184118254419041508, 15.47582110605087786463439853895, 16.78112915056809372972104844903, 18.96460454296572475775664511811, 20.02583321858768024715850251742
