| L(s) = 1 | + (0.256 − 0.222i)2-s + (−0.399 + 1.68i)3-s + (−0.268 + 1.86i)4-s + (0.197 − 0.432i)5-s + (0.272 + 0.521i)6-s + (0.641 − 2.18i)7-s + (0.713 + 1.11i)8-s + (−2.68 − 1.34i)9-s + (−0.0455 − 0.155i)10-s + (3.61 − 4.16i)11-s + (−3.03 − 1.19i)12-s + (−0.876 + 0.257i)13-s + (−0.321 − 0.703i)14-s + (0.650 + 0.505i)15-s + (−3.18 − 0.935i)16-s + (0.637 + 4.43i)17-s + ⋯ |
| L(s) = 1 | + (0.181 − 0.157i)2-s + (−0.230 + 0.973i)3-s + (−0.134 + 0.932i)4-s + (0.0883 − 0.193i)5-s + (0.111 + 0.213i)6-s + (0.242 − 0.825i)7-s + (0.252 + 0.392i)8-s + (−0.893 − 0.448i)9-s + (−0.0143 − 0.0490i)10-s + (1.08 − 1.25i)11-s + (−0.876 − 0.345i)12-s + (−0.243 + 0.0713i)13-s + (−0.0859 − 0.188i)14-s + (0.167 + 0.130i)15-s + (−0.796 − 0.233i)16-s + (0.154 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.860339 + 0.358164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.860339 + 0.358164i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.399 - 1.68i)T \) |
| 23 | \( 1 + (2.58 + 4.04i)T \) |
| good | 2 | \( 1 + (-0.256 + 0.222i)T + (0.284 - 1.97i)T^{2} \) |
| 5 | \( 1 + (-0.197 + 0.432i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.641 + 2.18i)T + (-5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-3.61 + 4.16i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.876 - 0.257i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.637 - 4.43i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (3.96 + 0.570i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (2.58 - 0.372i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.56 + 1.65i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.02 + 0.467i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (5.22 + 2.38i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.87 - 2.91i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 10.1iT - 47T^{2} \) |
| 53 | \( 1 + (6.20 + 1.82i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (1.57 + 5.37i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.10 + 3.27i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (7.57 - 6.56i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-9.28 + 8.04i)T + (10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.61 - 11.2i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (4.00 + 13.6i)T + (-66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.08 - 6.74i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-12.2 - 7.87i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.94 - 1.34i)T + (63.5 + 73.3i)T^{2} \) |
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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
−14.75354841844795667252315808676, −13.88755742930706376803972512533, −12.64086545387116446860555065974, −11.42805598966938317994168420065, −10.63838429943400705019211107465, −9.096880244108562291016779472619, −8.171718234197011385629429901320, −6.32563989714841073409681342595, −4.49387704677184885367986404987, −3.53634415005430285372586755603,
1.96030202608116459325118445370, 4.90932687470624797702058118621, 6.20102827276802758359328698960, 7.17882166756800826840366173805, 8.872110873502471411716713612555, 10.06869610519654476225962102890, 11.57961593140095975639302245590, 12.34699891620458097823201507711, 13.67504370481091440203828046158, 14.58056569786795276187417800988
