| L(s) = 1 | − 3.80i·2-s + 1.73·3-s − 10.4·4-s − 6.58i·6-s + (6.55 + 2.44i)7-s + 24.6i·8-s + 2.99·9-s + 14.4·11-s − 18.1·12-s + 16.9·13-s + (9.30 − 24.9i)14-s + 51.8·16-s − 13.0·17-s − 11.4i·18-s − 18.6i·19-s + ⋯ |
| L(s) = 1 | − 1.90i·2-s + 0.577·3-s − 2.61·4-s − 1.09i·6-s + (0.936 + 0.349i)7-s + 3.07i·8-s + 0.333·9-s + 1.31·11-s − 1.51·12-s + 1.30·13-s + (0.664 − 1.78i)14-s + 3.23·16-s − 0.765·17-s − 0.634i·18-s − 0.983i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.229646119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.229646119\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.55 - 2.44i)T \) |
| good | 2 | \( 1 + 3.80iT - 4T^{2} \) |
| 11 | \( 1 - 14.4T + 121T^{2} \) |
| 13 | \( 1 - 16.9T + 169T^{2} \) |
| 17 | \( 1 + 13.0T + 289T^{2} \) |
| 19 | \( 1 + 18.6iT - 361T^{2} \) |
| 23 | \( 1 + 10.3iT - 529T^{2} \) |
| 29 | \( 1 - 13.7T + 841T^{2} \) |
| 31 | \( 1 - 42.4iT - 961T^{2} \) |
| 37 | \( 1 - 28.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 5.84iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 81.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 35.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 68.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 47.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 47.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 125.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 129.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 42.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 25.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 28.7T + 9.40e3T^{2} \) |
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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
−10.57910324479559705184783509153, −9.457371420601088575302133873700, −8.707007177095441296426097628165, −8.403947346323794067809740650383, −6.62486854216743698984435634774, −5.00592005249835291143359237241, −4.17595676503585096733743774116, −3.24266551209601286034747743981, −2.02701062748546054784205604881, −1.11531779009844670739944088075,
1.28216616572477519646644852267, 3.87694228725931254941357721301, 4.35092207786375826173424804930, 5.73443420158410974597577114924, 6.47223644034592188244202547270, 7.44064664494938453566228111339, 8.176728773912727031328343649421, 8.841815443947649271596184295362, 9.535373761344019976086983722473, 10.80637881922236564326999632773
