| L(s) = 1 | + (0.655 − 1.27i)2-s + (−0.309 − 2.81i)3-s + (1.13 + 1.57i)4-s + (−3.99 − 1.87i)5-s + (−3.79 − 1.45i)6-s + (2.49 + 11.2i)7-s + (8.43 − 1.24i)8-s + (0.966 − 0.215i)9-s + (−5.01 + 3.87i)10-s + (−7.78 + 4.36i)11-s + (4.09 − 3.66i)12-s + (18.9 − 9.72i)13-s + (15.9 + 4.16i)14-s + (−4.03 + 11.8i)15-s + (1.43 − 4.21i)16-s + (4.96 + 8.84i)17-s + ⋯ |
| L(s) = 1 | + (0.327 − 0.637i)2-s + (−0.103 − 0.937i)3-s + (0.282 + 0.394i)4-s + (−0.799 − 0.374i)5-s + (−0.631 − 0.241i)6-s + (0.356 + 1.60i)7-s + (1.05 − 0.155i)8-s + (0.107 − 0.0239i)9-s + (−0.501 + 0.387i)10-s + (−0.707 + 0.397i)11-s + (0.341 − 0.305i)12-s + (1.45 − 0.748i)13-s + (1.13 + 0.297i)14-s + (−0.269 + 0.788i)15-s + (0.0899 − 0.263i)16-s + (0.291 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.00982 - 1.00925i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00982 - 1.00925i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 431 | \( 1 + (313. - 295. i)T \) |
| good | 2 | \( 1 + (-0.655 + 1.27i)T + (-2.32 - 3.25i)T^{2} \) |
| 3 | \( 1 + (0.309 + 2.81i)T + (-8.78 + 1.95i)T^{2} \) |
| 5 | \( 1 + (3.99 + 1.87i)T + (15.9 + 19.2i)T^{2} \) |
| 7 | \( 1 + (-2.49 - 11.2i)T + (-44.3 + 20.7i)T^{2} \) |
| 11 | \( 1 + (7.78 - 4.36i)T + (63.0 - 103. i)T^{2} \) |
| 13 | \( 1 + (-18.9 + 9.72i)T + (98.3 - 137. i)T^{2} \) |
| 17 | \( 1 + (-4.96 - 8.84i)T + (-150. + 246. i)T^{2} \) |
| 19 | \( 1 + (-23.6 + 11.0i)T + (230. - 277. i)T^{2} \) |
| 23 | \( 1 + (0.0572 + 0.0593i)T + (-19.3 + 528. i)T^{2} \) |
| 29 | \( 1 + (3.01 - 4.22i)T + (-271. - 795. i)T^{2} \) |
| 31 | \( 1 + (-8.48 - 22.1i)T + (-715. + 641. i)T^{2} \) |
| 37 | \( 1 + (-16.0 - 4.20i)T + (1.19e3 + 670. i)T^{2} \) |
| 41 | \( 1 + (17.0 - 49.9i)T + (-1.33e3 - 1.02e3i)T^{2} \) |
| 43 | \( 1 + (-30.6 + 39.6i)T + (-467. - 1.78e3i)T^{2} \) |
| 47 | \( 1 + (-5.84 + 79.8i)T + (-2.18e3 - 321. i)T^{2} \) |
| 53 | \( 1 + (31.4 - 2.30i)T + (2.77e3 - 408. i)T^{2} \) |
| 59 | \( 1 + (17.8 - 34.6i)T + (-2.02e3 - 2.83e3i)T^{2} \) |
| 61 | \( 1 + (-25.2 - 59.3i)T + (-2.58e3 + 2.67e3i)T^{2} \) |
| 67 | \( 1 + (-41.9 - 10.9i)T + (3.91e3 + 2.19e3i)T^{2} \) |
| 71 | \( 1 + (-5.78 + 79.0i)T + (-4.98e3 - 733. i)T^{2} \) |
| 73 | \( 1 + (-32.4 - 49.0i)T + (-2.08e3 + 4.90e3i)T^{2} \) |
| 79 | \( 1 + (-0.563 + 1.87i)T + (-5.20e3 - 3.44e3i)T^{2} \) |
| 83 | \( 1 + (27.0 - 2.97i)T + (6.72e3 - 1.49e3i)T^{2} \) |
| 89 | \( 1 + (45.7 - 8.44i)T + (7.39e3 - 2.82e3i)T^{2} \) |
| 97 | \( 1 + (-175. + 52.8i)T + (7.84e3 - 5.19e3i)T^{2} \) |
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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
−11.30249383086948403854533194677, −10.15486812796991951007780107848, −8.629683083065234740912829289843, −8.075723180138593860047502875847, −7.29237400532503913817910122881, −6.02151366123723915826499177017, −4.95193236338569908079369303210, −3.57039396364023630224419536229, −2.46491847098501666984979577588, −1.24762802872776713330514184796,
1.15891738718577041216107493835, 3.57784774904295415000032410493, 4.21992200847619011892767436596, 5.20646517341087110923733835250, 6.38162071102281087456244906017, 7.51797505351322667416812988050, 7.80992832771081798070466132773, 9.518425888897109157191242350004, 10.35580356778599808445011139929, 11.11498716096385543396195568249
