| L(s) = 1 | + (−0.178 − 1.40i)2-s + (1.23 + 0.450i)3-s + (−1.93 + 0.501i)4-s + (−0.503 − 2.85i)5-s + (0.410 − 1.81i)6-s + (2.71 + 1.56i)7-s + (1.04 + 2.62i)8-s + (−0.971 − 0.815i)9-s + (−3.91 + 1.21i)10-s + (−3.30 + 1.90i)11-s + (−2.61 − 0.251i)12-s + (1.53 + 4.21i)13-s + (1.71 − 4.08i)14-s + (0.661 − 3.75i)15-s + (3.49 − 1.94i)16-s + (0.0599 − 0.0503i)17-s + ⋯ |
| L(s) = 1 | + (−0.126 − 0.991i)2-s + (0.713 + 0.259i)3-s + (−0.968 + 0.250i)4-s + (−0.224 − 1.27i)5-s + (0.167 − 0.741i)6-s + (1.02 + 0.591i)7-s + (0.371 + 0.928i)8-s + (−0.323 − 0.271i)9-s + (−1.23 + 0.384i)10-s + (−0.996 + 0.575i)11-s + (−0.756 − 0.0724i)12-s + (0.425 + 1.16i)13-s + (0.457 − 1.09i)14-s + (0.170 − 0.969i)15-s + (0.874 − 0.485i)16-s + (0.0145 − 0.0122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.806369 - 0.557302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.806369 - 0.557302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.178 + 1.40i)T \) |
| 19 | \( 1 + (3.51 - 2.57i)T \) |
| good | 3 | \( 1 + (-1.23 - 0.450i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.503 + 2.85i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.71 - 1.56i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.30 - 1.90i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 4.21i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.0599 + 0.0503i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-2.21 - 0.390i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.332 + 0.395i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.35 + 2.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.4iT - 37T^{2} \) |
| 41 | \( 1 + (-0.203 + 0.558i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (10.9 - 1.92i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.45 - 1.73i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.929 - 0.163i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.93 + 5.82i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.585 - 3.32i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.61 - 1.35i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.503 - 2.85i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-13.7 - 4.99i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.53 - 2.37i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (4.69 + 2.71i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.01 - 2.79i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (6.08 + 7.25i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−14.19333852845704670279412740190, −13.02014743029390355331452593159, −12.11981027756681432000850386166, −11.17519055608402978349719056792, −9.624203690875081124038098642434, −8.671511536410141453219782190146, −8.148890440528879992727216194172, −5.21422502512314420693235549873, −4.12533665122196858604626447801, −2.07349093701478809585330490581,
3.14621968409691302582184654994, 5.14629827175807660017740575784, 6.76711126087070234963800170181, 7.940709191412332652137191489816, 8.378501799696395592966793487262, 10.38737510115031200188551572221, 11.03662015175294703523881335165, 13.24866444055552446165459928567, 13.84023750727518085029391955560, 14.88933631785831407867491067493
