L(s) = 1 | + (1.30 − 1.09i)2-s + (0.387 + 1.68i)3-s + (0.158 − 0.899i)4-s + (1.05 − 2.90i)5-s + (2.35 + 1.78i)6-s + (−0.421 − 0.730i)7-s + (0.927 + 1.60i)8-s + (−2.69 + 1.30i)9-s + (−1.80 − 4.95i)10-s + 5.25i·11-s + (1.58 − 0.0811i)12-s + (−2.06 − 5.67i)13-s + (−1.35 − 0.492i)14-s + (5.30 + 0.657i)15-s + (4.69 + 1.70i)16-s + (0.911 − 2.50i)17-s + ⋯ |
L(s) = 1 | + (0.924 − 0.775i)2-s + (0.223 + 0.974i)3-s + (0.0793 − 0.449i)4-s + (0.472 − 1.29i)5-s + (0.963 + 0.727i)6-s + (−0.159 − 0.275i)7-s + (0.327 + 0.567i)8-s + (−0.899 + 0.436i)9-s + (−0.569 − 1.56i)10-s + 1.58i·11-s + (0.456 − 0.0234i)12-s + (−0.572 − 1.57i)13-s + (−0.361 − 0.131i)14-s + (1.36 + 0.169i)15-s + (1.17 + 0.426i)16-s + (0.221 − 0.607i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.85687 - 0.408397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85687 - 0.408397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.387 - 1.68i)T \) |
| 19 | \( 1 + (3.91 - 1.92i)T \) |
good | 2 | \( 1 + (-1.30 + 1.09i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (-1.05 + 2.90i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.421 + 0.730i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 5.25iT - 11T^{2} \) |
| 13 | \( 1 + (2.06 + 5.67i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.911 + 2.50i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (5.64 + 0.995i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.889 - 5.04i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 - 4.15iT - 31T^{2} \) |
| 37 | \( 1 + 4.72iT - 37T^{2} \) |
| 41 | \( 1 + (2.20 - 1.85i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.757 + 4.29i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.55 - 0.979i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-9.57 - 8.03i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.518 + 2.94i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.29 + 2.29i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.11 - 1.33i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.22 + 4.38i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.376 + 2.13i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.27 + 3.51i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.69 + 4.44i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.29 - 7.33i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.81 - 2.16i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−12.51900302612433762044258752640, −12.18672899857493034458516226954, −10.51326152878653930682840436893, −10.03485520792213490620331687076, −8.848901479149881266093388426282, −7.73487496751798895406095028563, −5.44866045564117867714965845889, −4.86114452386002843752406440360, −3.84889817338489063272967499478, −2.27731357854904212441824830815,
2.37380680424676198446337872767, 3.84940659555384118565077710424, 5.87511383702001386205943559588, 6.32788918767248900796750996106, 7.14269203568923450247201593839, 8.405616909986732949689092205811, 9.808866462075357779346346096148, 11.10540810240802168001157270310, 12.04062359313350450017431844843, 13.36990522672707916366551368063