L(s) = 1 | + (−0.944 − 1.63i)2-s + (−0.783 − 1.35i)3-s + (2.21 − 3.83i)4-s + 2.20·5-s + (−1.47 + 2.56i)6-s + (0.902 − 1.56i)7-s − 23.4·8-s + (12.2 − 21.2i)9-s + (−2.08 − 3.60i)10-s + (−5.5 − 9.52i)11-s − 6.94·12-s + (22.7 − 40.9i)13-s − 3.40·14-s + (−1.72 − 2.99i)15-s + (4.45 + 7.71i)16-s + (−49.5 + 85.8i)17-s + ⋯ |
L(s) = 1 | + (−0.333 − 0.578i)2-s + (−0.150 − 0.261i)3-s + (0.276 − 0.479i)4-s + 0.197·5-s + (−0.100 + 0.174i)6-s + (0.0487 − 0.0843i)7-s − 1.03·8-s + (0.454 − 0.787i)9-s + (−0.0658 − 0.114i)10-s + (−0.150 − 0.261i)11-s − 0.166·12-s + (0.486 − 0.873i)13-s − 0.0650·14-s + (−0.0297 − 0.0515i)15-s + (0.0696 + 0.120i)16-s + (−0.707 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.162117 - 1.13364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.162117 - 1.13364i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (5.5 + 9.52i)T \) |
| 13 | \( 1 + (-22.7 + 40.9i)T \) |
good | 2 | \( 1 + (0.944 + 1.63i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (0.783 + 1.35i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 - 2.20T + 125T^{2} \) |
| 7 | \( 1 + (-0.902 + 1.56i)T + (-171.5 - 297. i)T^{2} \) |
| 17 | \( 1 + (49.5 - 85.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (8.59 - 14.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (78.0 + 135. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (25.3 + 43.9i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 35.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + (93.9 + 162. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-140. - 242. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-152. + 263. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 314.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 8.37T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-111. + 193. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-344. + 596. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-89.4 - 154. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-513. + 890. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 - 725.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 377.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 705.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-521. - 903. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-700. + 1.21e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.17988297852095146509780464080, −10.95629705094971055374600943862, −10.33105665166690870422911118715, −9.268707636135670996499655422415, −8.099410738375898969306138203930, −6.49434836631951447904183877475, −5.81201864220388064744571455614, −3.84112727530441934047735720871, −2.11204179948638766829385705452, −0.61988189035350284793164679213,
2.24318008313697639378765624917, 4.06701536241619142765871344653, 5.52968629918946175019364992303, 6.86963246753946992348809086720, 7.68348176251330569572969731001, 8.894962480155024572452636028559, 9.825400762458810174417020427453, 11.19736371151117974741308315426, 11.90868826362819352430398469863, 13.23895691639903596861422379360