| L(s) = 1 | + (0.373 − 1.96i)2-s + (−1.67 − 0.448i)3-s + (−3.72 − 1.46i)4-s + (−3.18 + 0.853i)5-s + (−1.50 + 3.11i)6-s + (−6.82 + 1.54i)7-s + (−4.27 + 6.76i)8-s + (2.59 + 1.50i)9-s + (0.486 + 6.57i)10-s + (2.56 + 0.687i)11-s + (5.56 + 4.12i)12-s + (12.7 − 12.7i)13-s + (0.478 + 13.9i)14-s + 5.71·15-s + (11.6 + 10.9i)16-s + (5.63 − 3.25i)17-s + ⋯ |
| L(s) = 1 | + (0.186 − 0.982i)2-s + (−0.557 − 0.149i)3-s + (−0.930 − 0.367i)4-s + (−0.637 + 0.170i)5-s + (−0.250 + 0.519i)6-s + (−0.975 + 0.220i)7-s + (−0.534 + 0.845i)8-s + (0.288 + 0.166i)9-s + (0.0486 + 0.657i)10-s + (0.233 + 0.0625i)11-s + (0.463 + 0.343i)12-s + (0.979 − 0.979i)13-s + (0.0341 + 0.999i)14-s + 0.380·15-s + (0.730 + 0.682i)16-s + (0.331 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00180i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.819052 - 0.000739959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.819052 - 0.000739959i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.373 + 1.96i)T \) |
| 3 | \( 1 + (1.67 + 0.448i)T \) |
| 7 | \( 1 + (6.82 - 1.54i)T \) |
| good | 5 | \( 1 + (3.18 - 0.853i)T + (21.6 - 12.5i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 0.687i)T + (104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (-12.7 + 12.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (-5.63 + 3.25i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-7.08 - 26.4i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (-21.6 - 12.4i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (5.74 + 5.74i)T + 841iT^{2} \) |
| 31 | \( 1 + (-8.36 + 4.83i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-7.97 - 29.7i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 49.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (52.0 - 52.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-39.8 - 23.0i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-96.4 - 25.8i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-8.57 + 32.0i)T + (-3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 38.5i)T + (-3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (31.7 - 118. i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 2.45iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (51.8 + 89.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (26.2 - 45.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (11.0 - 11.0i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-78.5 + 136. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 40.5iT - 9.40e3T^{2} \) |
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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
−11.59276182889649946464430322017, −10.44312445060385093184415319709, −9.865972736971450097194584106241, −8.685513941023250442259557977120, −7.61617736466152361229890143532, −6.19009438975475626522788351651, −5.38438886448433996235667696595, −3.85813852426465291901894818943, −3.10687892118244907164529713890, −1.13471053658122714176577681918,
0.48231272626138913744909959317, 3.48095514629293953202711192491, 4.34093775279071674044901708893, 5.52427576099535734756760658900, 6.66657696172843155116634071573, 7.10431054466118603591243423550, 8.565348385518623934956782699701, 9.206772974519595315262138410508, 10.32541569644060507279151549455, 11.50372628916277487744721753584
