L(s) = 1 | − 5.64·2-s − 9·3-s − 0.188·4-s + 25·5-s + 50.7·6-s + 145.·7-s + 181.·8-s + 81·9-s − 141.·10-s − 121·11-s + 1.69·12-s − 69.9·13-s − 817.·14-s − 225·15-s − 1.01e3·16-s − 500.·17-s − 456.·18-s − 670.·19-s − 4.70·20-s − 1.30e3·21-s + 682.·22-s + 791.·23-s − 1.63e3·24-s + 625·25-s + 394.·26-s − 729·27-s − 27.3·28-s + ⋯ |
L(s) = 1 | − 0.997·2-s − 0.577·3-s − 0.00588·4-s + 0.447·5-s + 0.575·6-s + 1.11·7-s + 1.00·8-s + 0.333·9-s − 0.445·10-s − 0.301·11-s + 0.00339·12-s − 0.114·13-s − 1.11·14-s − 0.258·15-s − 0.994·16-s − 0.420·17-s − 0.332·18-s − 0.426·19-s − 0.00263·20-s − 0.645·21-s + 0.300·22-s + 0.311·23-s − 0.579·24-s + 0.200·25-s + 0.114·26-s − 0.192·27-s − 0.00658·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9596889671\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9596889671\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 + 5.64T + 32T^{2} \) |
| 7 | \( 1 - 145.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 69.9T + 3.71e5T^{2} \) |
| 17 | \( 1 + 500.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 670.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 791.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.68e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.16e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.88e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.65e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.54e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.93e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.09e4T + 8.58e9T^{2} \) |
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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
−11.57312022533148106213853824767, −10.74842132224563290986945716651, −9.972486242375844595378535073880, −8.830046302966854482740088656169, −7.994437793409911235673916520874, −6.85529341403178437414155954229, −5.35296769742148392019876554355, −4.41737165682017092840881371958, −2.03549059882497724123099092593, −0.77937066771802246596478905562,
0.77937066771802246596478905562, 2.03549059882497724123099092593, 4.41737165682017092840881371958, 5.35296769742148392019876554355, 6.85529341403178437414155954229, 7.994437793409911235673916520874, 8.830046302966854482740088656169, 9.972486242375844595378535073880, 10.74842132224563290986945716651, 11.57312022533148106213853824767