| L(s) = 1 | − 1.41·2-s + 2.82·3-s + 2.00·4-s − 4.00·6-s + 5i·7-s − 2.82·8-s − 0.999·9-s + 5·11-s + 5.65·12-s − 16.9·13-s − 7.07i·14-s + 4.00·16-s − 25i·17-s + 1.41·18-s − 19·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.942·3-s + 0.500·4-s − 0.666·6-s + 0.714i·7-s − 0.353·8-s − 0.111·9-s + 0.454·11-s + 0.471·12-s − 1.30·13-s − 0.505i·14-s + 0.250·16-s − 1.47i·17-s + 0.0785·18-s − 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8104160266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8104160266\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 2.82T + 9T^{2} \) |
| 7 | \( 1 - 5iT - 49T^{2} \) |
| 11 | \( 1 - 5T + 121T^{2} \) |
| 13 | \( 1 + 16.9T + 169T^{2} \) |
| 17 | \( 1 + 25iT - 289T^{2} \) |
| 23 | \( 1 - 10iT - 529T^{2} \) |
| 29 | \( 1 + 42.4iT - 841T^{2} \) |
| 31 | \( 1 + 42.4iT - 961T^{2} \) |
| 37 | \( 1 - 25.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 42.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 95T + 3.72e3T^{2} \) |
| 67 | \( 1 + 110.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 25iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 42.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 130iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 16.9T + 9.40e3T^{2} \) |
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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
−9.448966994660946019870848090287, −8.880697513565207717876414196221, −7.970637994403256061307550248404, −7.38400479752541540870721176559, −6.29547085421051637320616782296, −5.27398323550185810298959825925, −4.00113483016410074788232058266, −2.60937850128345445730100158244, −2.25270194827171306605159112695, −0.27018765032199436040972925256,
1.46939564708239086086383395371, 2.58405898547780088613379905461, 3.61007751045045512311112835383, 4.66191448052582623959753830694, 6.08583799310987843283751019999, 7.00333731858846004379330917007, 7.74128469561995849586005530143, 8.608030800877563708999649716200, 9.030388256652598746085271916968, 10.17227191187478884312350446275
