L(s) = 1 | + (−1.5 + 0.866i)3-s + (−3 − 5.19i)5-s + 5·7-s + (1.5 − 2.59i)9-s + (−16.5 − 9.52i)13-s + (9 + 5.19i)15-s + (−3 − 5.19i)17-s + (13 − 13.8i)19-s + (−7.5 + 4.33i)21-s + (−12 + 20.7i)23-s + (−5.5 + 9.52i)25-s + 5.19i·27-s + (27 + 15.5i)29-s + 29.4i·31-s + (−15 − 25.9i)35-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.288i)3-s + (−0.600 − 1.03i)5-s + 0.714·7-s + (0.166 − 0.288i)9-s + (−1.26 − 0.732i)13-s + (0.599 + 0.346i)15-s + (−0.176 − 0.305i)17-s + (0.684 − 0.729i)19-s + (−0.357 + 0.206i)21-s + (−0.521 + 0.903i)23-s + (−0.220 + 0.381i)25-s + 0.192i·27-s + (0.931 + 0.537i)29-s + 0.949i·31-s + (−0.428 − 0.742i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1542491974\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1542491974\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 19 | \( 1 + (-13 + 13.8i)T \) |
good | 5 | \( 1 + (3 + 5.19i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 - 5T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + (16.5 + 9.52i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (12 - 20.7i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-27 - 15.5i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 29.4iT - 961T^{2} \) |
| 37 | \( 1 + 60.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (36 - 20.7i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (12.5 + 21.6i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (21 - 36.3i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-54 - 31.1i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (63 - 36.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (21.5 - 37.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (49.5 + 28.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (54 - 31.1i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 126T + 6.88e3T^{2} \) |
| 89 | \( 1 + (9 + 5.19i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (114 - 65.8i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.402694254736172943927003576163, −8.622889102468590100163978499284, −7.73774751553201633396306582281, −7.06158990459906161662017566222, −5.57259733977194787694948563650, −4.96793362982337263788817556352, −4.34712322953427252703238614140, −2.95630584492293215192943945633, −1.29637605717862797527193283291, −0.05610037965104912722668588883,
1.76791714782123571490302283353, 2.92004431565720049781975229487, 4.21938234099063058897154533427, 5.02324834279201588034082146053, 6.26513088224085724924564731362, 6.95128026036942722069777889816, 7.73657486242439799146546842057, 8.408258494524091025317623760409, 9.854031842611033756411292585747, 10.32665792727635543057324546143