L(s) = 1 | + (1.80 + 1.31i)3-s − 0.381·5-s + (0.0729 − 0.224i)7-s + (0.618 + 1.90i)9-s + (−0.381 + 1.17i)11-s + (−3.73 − 2.71i)13-s + (−0.690 − 0.502i)15-s + (0.690 + 2.12i)17-s + (5.42 − 3.94i)19-s + (0.427 − 0.310i)21-s + (−0.545 − 1.67i)23-s − 4.85·25-s + (0.690 − 2.12i)27-s + (−4.54 + 3.30i)29-s + (−4.19 − 3.66i)31-s + ⋯ |
L(s) = 1 | + (1.04 + 0.758i)3-s − 0.170·5-s + (0.0275 − 0.0848i)7-s + (0.206 + 0.634i)9-s + (−0.115 + 0.354i)11-s + (−1.03 − 0.752i)13-s + (−0.178 − 0.129i)15-s + (0.167 + 0.515i)17-s + (1.24 − 0.904i)19-s + (0.0931 − 0.0677i)21-s + (−0.113 − 0.349i)23-s − 0.970·25-s + (0.132 − 0.409i)27-s + (−0.844 + 0.613i)29-s + (−0.752 − 0.658i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29324 + 0.366877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29324 + 0.366877i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 31 | \( 1 + (4.19 + 3.66i)T \) |
good | 3 | \( 1 + (-1.80 - 1.31i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 7 | \( 1 + (-0.0729 + 0.224i)T + (-5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (0.381 - 1.17i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (3.73 + 2.71i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.690 - 2.12i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-5.42 + 3.94i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.545 + 1.67i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (4.54 - 3.30i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 + (8.47 - 6.15i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (0.927 - 0.673i)T + (13.2 - 40.8i)T^{2} \) |
| 47 | \( 1 + (-6.92 - 5.03i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.83 + 8.73i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.2 - 7.46i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 + (-4.04 - 12.4i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.0450 + 0.138i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (6.35 - 4.61i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.95 - 15.2i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.45 + 4.47i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−13.65637726517266538816719608595, −12.60764058769592457123966377202, −11.36737021361213843571845508070, −10.02623521330768693392755135103, −9.458338073601617187507314253927, −8.214914697319898431383017073307, −7.27647888124592782538402290975, −5.35639575628121411714866196981, −3.99473777609185370911692397321, −2.70678949281703143112566840538,
2.07643553086583651340959732352, 3.56935255819313689782366702451, 5.43838059902636109936904563161, 7.16878497280932737513455594618, 7.79151135022666961191489077567, 8.983237240510302379593111486059, 9.931015782936032041508269101889, 11.55891560241883509095607332788, 12.34221357214069532668881736800, 13.60318016566813629747387917755