L(s) = 1 | − 2.15i·2-s − 5.19·3-s + 11.3·4-s + 30.9·5-s + 11.2i·6-s + 73.3·7-s − 58.9i·8-s + 27·9-s − 66.6i·10-s + 203. i·11-s − 58.9·12-s + 115. i·13-s − 158. i·14-s − 160.·15-s + 54.5·16-s − 149.·17-s + ⋯ |
L(s) = 1 | − 0.538i·2-s − 0.577·3-s + 0.709·4-s + 1.23·5-s + 0.311i·6-s + 1.49·7-s − 0.921i·8-s + 0.333·9-s − 0.666i·10-s + 1.68i·11-s − 0.409·12-s + 0.681i·13-s − 0.806i·14-s − 0.714·15-s + 0.213·16-s − 0.517·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.725010205\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.725010205\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (-3.34e3 - 964. i)T \) |
good | 2 | \( 1 + 2.15iT - 16T^{2} \) |
| 5 | \( 1 - 30.9T + 625T^{2} \) |
| 7 | \( 1 - 73.3T + 2.40e3T^{2} \) |
| 11 | \( 1 - 203. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 115. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 149.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 97.5T + 1.30e5T^{2} \) |
| 23 | \( 1 - 785. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 279.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 520. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 737. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 540.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.17e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.50e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.63e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 5.34e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 1.69e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 9.06e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 876. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 7.89e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.53e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.14e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.51e4iT - 8.85e7T^{2} \) |
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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
−11.77861283691194148030354540763, −11.09175147366416862512834900494, −10.12954707957475856401990599609, −9.353880462393328705419348937879, −7.61108269463670313010450222958, −6.68697704716295923817858442816, −5.46308779297468250210203529201, −4.36376629531641198804428574897, −2.03202839508454615828539318389, −1.66494174373677407925847717187,
1.23023551475726506651298933362, 2.58121780474787656346136911357, 4.89564772815068459986381528897, 5.81702695992036129728186742574, 6.47518533040311232712689086671, 7.953996759679657297707328257392, 8.723155067321634859949499126802, 10.47693301141330455760702510958, 10.93814085320820012144750916385, 11.82344409771297229678444571040