| 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
−11.50372628916277487744721753584, −10.32541569644060507279151549455, −9.206772974519595315262138410508, −8.565348385518623934956782699701, −7.10431054466118603591243423550, −6.66657696172843155116634071573, −5.52427576099535734756760658900, −4.34093775279071674044901708893, −3.48095514629293953202711192491, −0.48231272626138913744909959317,
1.13471053658122714176577681918, 3.10687892118244907164529713890, 3.85813852426465291901894818943, 5.38438886448433996235667696595, 6.19009438975475626522788351651, 7.61617736466152361229890143532, 8.685513941023250442259557977120, 9.865972736971450097194584106241, 10.44312445060385093184415319709, 11.59276182889649946464430322017
