| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (−1.38 + 4.80i)5-s + (−5.24 + 4.63i)7-s + 2.82i·8-s + (6.79 + 1.95i)10-s − 11.7·11-s + 24.8·13-s + (6.54 + 7.42i)14-s + 4.00·16-s − 7.26·17-s − 23.0i·19-s + (2.76 − 9.61i)20-s + 16.6i·22-s − 26.4i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + (−0.276 + 0.961i)5-s + (−0.749 + 0.661i)7-s + 0.353i·8-s + (0.679 + 0.195i)10-s − 1.06·11-s + 1.90·13-s + (0.467 + 0.530i)14-s + 0.250·16-s − 0.427·17-s − 1.21i·19-s + (0.138 − 0.480i)20-s + 0.756i·22-s − 1.14i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4397920237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4397920237\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.38 - 4.80i)T \) |
| 7 | \( 1 + (5.24 - 4.63i)T \) |
| good | 11 | \( 1 + 11.7T + 121T^{2} \) |
| 13 | \( 1 - 24.8T + 169T^{2} \) |
| 17 | \( 1 + 7.26T + 289T^{2} \) |
| 19 | \( 1 + 23.0iT - 361T^{2} \) |
| 23 | \( 1 + 26.4iT - 529T^{2} \) |
| 29 | \( 1 + 57.0T + 841T^{2} \) |
| 31 | \( 1 - 10.2iT - 961T^{2} \) |
| 37 | \( 1 - 14.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 51.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 64.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 22.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 91.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 71.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 45.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 17.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 77.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 6.15T + 9.40e3T^{2} \) |
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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
−10.35889815260174873848075668076, −9.111803438308832095546899160940, −8.564632323201687168414184001896, −7.33651555928819987518423335116, −6.34886053504670908541112835906, −5.48903054741281673033428458342, −4.01271436637180874749932935267, −3.10821768676822351111551601740, −2.23342381349774551922800508623, −0.17034135538824398539981172710,
1.32597433553779167042715615516, 3.51536459702572872924163666050, 4.19279931806428823877675576505, 5.56453313659894623039100258026, 6.07488860912294585494190721997, 7.43517068744612493925540378967, 8.005072069750520009301452228728, 8.926684806311338904984400494439, 9.686929679361568650113813545024, 10.69033093813693234398439276607
