L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s − 230.·7-s − 64·8-s + 81·9-s + 100·10-s + 628.·11-s − 144·12-s + 803.·13-s + 922.·14-s + 225·15-s + 256·16-s − 802.·17-s − 324·18-s + 361·19-s − 400·20-s + 2.07e3·21-s − 2.51e3·22-s − 2.59e3·23-s + 576·24-s + 625·25-s − 3.21e3·26-s − 729·27-s − 3.69e3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s − 1.77·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.56·11-s − 0.288·12-s + 1.31·13-s + 1.25·14-s + 0.258·15-s + 0.250·16-s − 0.673·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 1.02·21-s − 1.10·22-s − 1.02·23-s + 0.204·24-s + 0.200·25-s − 0.932·26-s − 0.192·27-s − 0.889·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5920156993\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5920156993\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
good | 7 | \( 1 + 230.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 628.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 803.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 802.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.31e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.48e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.15e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.94e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.98e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.31e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.07e5T + 8.58e9T^{2} \) |
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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
−9.841693010553907654034559691786, −9.185225257437353599303938031500, −8.400530401296270714192716436250, −6.93602464989096327073657692022, −6.58156873357932077037621243691, −5.76302336806073178306341762779, −3.95771650768883332522073245765, −3.40291540628465690194466327935, −1.64980369824410100090104980436, −0.43808037305316715281099766231,
0.43808037305316715281099766231, 1.64980369824410100090104980436, 3.40291540628465690194466327935, 3.95771650768883332522073245765, 5.76302336806073178306341762779, 6.58156873357932077037621243691, 6.93602464989096327073657692022, 8.400530401296270714192716436250, 9.185225257437353599303938031500, 9.841693010553907654034559691786