| L(s) = 1 | + (2.88 + 4.32i)3-s + (−0.711 + 3.57i)5-s + (0.644 + 3.23i)7-s + (−6.88 + 16.6i)9-s + (−2.02 − 1.35i)11-s + (−7.73 − 7.73i)13-s + (−17.5 + 7.24i)15-s + (15.0 − 7.93i)17-s + (3.50 + 8.45i)19-s + (−12.1 + 12.1i)21-s + (1.91 − 2.87i)23-s + (10.8 + 4.47i)25-s + (−45.8 + 9.12i)27-s + (−23.5 − 4.69i)29-s + (23.0 − 15.3i)31-s + ⋯ |
| L(s) = 1 | + (0.962 + 1.44i)3-s + (−0.142 + 0.715i)5-s + (0.0920 + 0.462i)7-s + (−0.765 + 1.84i)9-s + (−0.183 − 0.122i)11-s + (−0.594 − 0.594i)13-s + (−1.16 + 0.483i)15-s + (0.884 − 0.466i)17-s + (0.184 + 0.445i)19-s + (−0.577 + 0.577i)21-s + (0.0834 − 0.124i)23-s + (0.432 + 0.179i)25-s + (−1.69 + 0.337i)27-s + (−0.813 − 0.161i)29-s + (0.742 − 0.496i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.826424 + 1.80905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.826424 + 1.80905i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-15.0 + 7.93i)T \) |
| good | 3 | \( 1 + (-2.88 - 4.32i)T + (-3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (0.711 - 3.57i)T + (-23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-0.644 - 3.23i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (2.02 + 1.35i)T + (46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (7.73 + 7.73i)T + 169iT^{2} \) |
| 19 | \( 1 + (-3.50 - 8.45i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-1.91 + 2.87i)T + (-202. - 488. i)T^{2} \) |
| 29 | \( 1 + (23.5 + 4.69i)T + (776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-23.0 + 15.3i)T + (367. - 887. i)T^{2} \) |
| 37 | \( 1 + (21.3 + 32.0i)T + (-523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-15.0 - 75.4i)T + (-1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (20.7 - 50.0i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-5.13 - 5.13i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-21.8 - 52.6i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-21.2 - 8.82i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-40.6 + 8.09i)T + (3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 17.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-28.6 - 42.8i)T + (-1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-12.1 + 60.9i)T + (-4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (98.5 + 65.8i)T + (2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-86.9 + 36.0i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-19.6 + 19.6i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-14.9 - 2.97i)T + (8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.80399127067568172400208385186, −10.80656410528141208453142691743, −10.02479484872120356452329531599, −9.371894085117248262768308652506, −8.278523904167639701601163205902, −7.45905174794985585567222915811, −5.73561432859646158739645904092, −4.67928255685328590214112082675, −3.38529248067098010663625613811, −2.64949477822377885317259334153,
0.945483269325713554188446850939, 2.22264187964249293159604152325, 3.68490486811240465734832196120, 5.23169416876632159042839238612, 6.77741561423272211425915391774, 7.42902294733338498938785886245, 8.370737983149478395962735854680, 9.061619556694008872822234636588, 10.28007492935350254373435124930, 11.80532142025728355018810689391
