| L(s) = 1 | + (−0.623 + 0.781i)2-s + (0.792 − 2.01i)3-s + (−0.222 − 0.974i)4-s + (0.0747 + 0.997i)5-s + (1.08 + 1.87i)6-s + (−2.30 + 3.99i)7-s + (0.900 + 0.433i)8-s + (−1.24 − 1.15i)9-s + (−0.826 − 0.563i)10-s + (−1.21 + 5.32i)11-s + (−2.14 − 0.323i)12-s + (−1.56 + 1.07i)13-s + (−1.68 − 4.29i)14-s + (2.07 + 0.639i)15-s + (−0.900 + 0.433i)16-s + (0.0194 − 0.259i)17-s + ⋯ |
| L(s) = 1 | + (−0.440 + 0.552i)2-s + (0.457 − 1.16i)3-s + (−0.111 − 0.487i)4-s + (0.0334 + 0.445i)5-s + (0.442 + 0.766i)6-s + (−0.871 + 1.51i)7-s + (0.318 + 0.153i)8-s + (−0.416 − 0.386i)9-s + (−0.261 − 0.178i)10-s + (−0.366 + 1.60i)11-s + (−0.619 − 0.0933i)12-s + (−0.435 + 0.296i)13-s + (−0.450 − 1.14i)14-s + (0.535 + 0.165i)15-s + (−0.225 + 0.108i)16-s + (0.00471 − 0.0629i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00732 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00732 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.673854 + 0.678809i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.673854 + 0.678809i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.623 - 0.781i)T \) |
| 5 | \( 1 + (-0.0747 - 0.997i)T \) |
| 43 | \( 1 + (2.33 - 6.12i)T \) |
| good | 3 | \( 1 + (-0.792 + 2.01i)T + (-2.19 - 2.04i)T^{2} \) |
| 7 | \( 1 + (2.30 - 3.99i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.21 - 5.32i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (1.56 - 1.07i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.0194 + 0.259i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 1.49i)T + (1.41 - 18.9i)T^{2} \) |
| 23 | \( 1 + (1.16 - 0.358i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-1.62 - 4.14i)T + (-21.2 + 19.7i)T^{2} \) |
| 31 | \( 1 + (-3.11 - 0.469i)T + (29.6 + 9.13i)T^{2} \) |
| 37 | \( 1 + (1.30 + 2.26i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.48 + 5.62i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (-1.64 - 7.18i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (8.94 + 6.09i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-9.17 + 4.41i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (5.62 - 0.847i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (6.69 - 6.21i)T + (5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-12.5 - 3.86i)T + (58.6 + 39.9i)T^{2} \) |
| 73 | \( 1 + (7.04 - 4.80i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (5.34 - 9.25i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.70 + 11.9i)T + (-60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (3.47 - 8.86i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-2.23 + 9.79i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−11.59315175632626492731605528168, −10.09859040810194052204522717561, −9.457639305086941596474655365462, −8.562195962445423430843570641550, −7.51616209725661214210284730895, −6.92923898933524564923077725484, −6.10634364104171074439275048255, −4.89703788330098400213880430892, −2.79281563188944988860385526640, −1.97588507403210412675145097978,
0.65994889416632295701455763689, 3.07212333007465496685562252886, 3.72425389708116125242257231917, 4.70734681103004795394102236982, 6.17410764546876766202797237776, 7.56595480853465750927575978499, 8.450858874406587158827493390234, 9.361244765876648652904165061971, 10.17806849362109014658045137747, 10.48288212386139850173092002164
