L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + 5·11-s + (1.73 − i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 3i·18-s + (0.5 − 4.33i)19-s + (4.33 + 2.5i)22-s + (0.866 − 0.5i)23-s + 1.99·26-s + (0.866 − 0.499i)28-s + (3 + 5.19i)29-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.377i·7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + 1.50·11-s + (0.480 − 0.277i)13-s + (0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.707i·18-s + (0.114 − 0.993i)19-s + (0.923 + 0.533i)22-s + (0.180 − 0.104i)23-s + 0.392·26-s + (0.163 − 0.0944i)28-s + (0.557 + 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39643 + 0.0454671i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39643 + 0.0454671i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.5 + 4.33i)T \) |
good | 3 | \( 1 + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 11iT - 37T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.19 + 3i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.33 + 2.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (12.1 + 7i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.73 + i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.03817646324843222578604389373, −8.941044483452758945288194564106, −8.561148318474120406395712897310, −7.15726394075252059024665525902, −6.62924758859714427625287195375, −5.84621031809005544773727977612, −4.68429170050238004615634909356, −3.78491720508854624424084559504, −2.94110387727781576997642837003, −1.11019688924150693909677833388,
1.43388620901607753058240750596, 2.59929520428756840606197565197, 3.79319029918689674980888800524, 4.59143986355492498312811661653, 5.79376537031522619107026680369, 6.28336990482225290358656533672, 7.46663885377775094129555481544, 8.461627883055480396664624415690, 9.245661705474416673785142636780, 10.13895998404196928474934498575