L(s) = 1 | + (0.713 − 0.259i)2-s + (0.884 − 1.48i)3-s + (−1.09 + 0.915i)4-s + (0.125 − 0.713i)5-s + (0.244 − 1.29i)6-s + (2.02 − 3.50i)7-s + (−1.29 + 2.25i)8-s + (−1.43 − 2.63i)9-s + (−0.0954 − 0.541i)10-s + 2.48·11-s + (0.397 + 2.43i)12-s + (0.692 + 3.93i)13-s + (0.533 − 3.02i)14-s + (−0.950 − 0.818i)15-s + (0.151 − 0.861i)16-s + (−1.06 + 6.06i)17-s + ⋯ |
L(s) = 1 | + (0.504 − 0.183i)2-s + (0.510 − 0.859i)3-s + (−0.545 + 0.457i)4-s + (0.0562 − 0.319i)5-s + (0.0998 − 0.527i)6-s + (0.764 − 1.32i)7-s + (−0.459 + 0.795i)8-s + (−0.478 − 0.878i)9-s + (−0.0301 − 0.171i)10-s + 0.748·11-s + (0.114 + 0.702i)12-s + (0.192 + 1.08i)13-s + (0.142 − 0.808i)14-s + (−0.245 − 0.211i)15-s + (0.0379 − 0.215i)16-s + (−0.259 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36366 - 0.755142i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36366 - 0.755142i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.884 + 1.48i)T \) |
| 19 | \( 1 + (3.54 + 2.53i)T \) |
good | 2 | \( 1 + (-0.713 + 0.259i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.125 + 0.713i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.02 + 3.50i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 + (-0.692 - 3.93i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.06 - 6.06i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (2.82 - 2.36i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (5.97 - 5.01i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 - 8.27T + 37T^{2} \) |
| 41 | \( 1 + (1.35 - 0.494i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.10 - 0.927i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.55 + 1.30i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (10.0 + 3.65i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-8.12 - 6.81i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (1.26 + 7.17i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.28 + 2.28i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.157 + 0.0572i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (7.43 + 6.24i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.362 - 2.05i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.16 + 12.4i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.93 + 4.14i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.38 + 3.41i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.94354148466234778845668059717, −11.81394053013314765407059556701, −10.95186490698559185425901277762, −9.229621550674132378110873461051, −8.467898593956651067888034098135, −7.48971162021812466219600922277, −6.33945905829660953528923973377, −4.50558088158472650558776041481, −3.71988605684258376139085304796, −1.64851054875462439602630258741,
2.64176933757425952214787661871, 4.19870562125488427576812682081, 5.22443139945238279628060372501, 6.13665727239603115053435613347, 8.066260030877079866343819808941, 8.976581572347854885007176206131, 9.752554275848289464643717032798, 10.87253876143974334018683170141, 11.94537164041571940609530456086, 13.13638051491401526225187029970