L(s) = 1 | − 13.1·2-s + 15.5i·3-s + 109.·4-s + 79.5i·5-s − 205. i·6-s − 601.·8-s − 243·9-s − 1.04e3i·10-s + 822.·11-s + 1.70e3i·12-s + 2.42e3i·13-s − 1.24e3·15-s + 913.·16-s + 7.79e3i·17-s + 3.20e3·18-s − 6.68e3i·19-s + ⋯ |
L(s) = 1 | − 1.64·2-s + 0.577i·3-s + 1.71·4-s + 0.636i·5-s − 0.951i·6-s − 1.17·8-s − 0.333·9-s − 1.04i·10-s + 0.617·11-s + 0.989i·12-s + 1.10i·13-s − 0.367·15-s + 0.223·16-s + 1.58i·17-s + 0.549·18-s − 0.974i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8188133939\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8188133939\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 15.5iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 13.1T + 64T^{2} \) |
| 5 | \( 1 - 79.5iT - 1.56e4T^{2} \) |
| 11 | \( 1 - 822.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 2.42e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 7.79e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 6.68e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.88e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 1.38e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.78e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 7.96e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.91e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.18e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 5.01e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.86e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.25e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.44e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 2.35e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 9.62e4T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.75e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 6.81e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.28e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.72e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 6.20e5iT - 8.32e11T^{2} \) |
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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
−11.59656378146180161437561671056, −11.01797942526577519045005852243, −10.12164970197107064672563149925, −9.187350098992822755628336890492, −8.501657367656905142092228460726, −7.13385324657553115067452288922, −6.34053773972021615641053671251, −4.33567592810404601831685662055, −2.67194511566084456211540819502, −1.18780453035651825346332890748,
0.54221659349303010969648061742, 1.27437456335060212881789862430, 2.86835903143948128891538171692, 5.11580960274797517862985327849, 6.66540427706292542569006753684, 7.57599449978181353800684500579, 8.537182120657453174974005233030, 9.270678684099442513311577338819, 10.31956488114792054417417528968, 11.38731960545699456357988652706