| L(s) = 1 | + (0.593 − 0.804i)2-s + (2.09 − 0.396i)3-s + (−0.294 − 0.955i)4-s + (1.87 + 1.22i)5-s + (0.926 − 1.92i)6-s + (1.00 − 2.44i)7-s + (−0.943 − 0.330i)8-s + (1.44 − 0.568i)9-s + (2.09 − 0.782i)10-s + (−0.581 + 1.48i)11-s + (−0.997 − 1.88i)12-s + (−1.68 − 0.190i)13-s + (−1.37 − 2.26i)14-s + (4.41 + 1.81i)15-s + (−0.826 + 0.563i)16-s + (−0.461 − 0.0172i)17-s + ⋯ |
| L(s) = 1 | + (0.419 − 0.568i)2-s + (1.21 − 0.229i)3-s + (−0.147 − 0.477i)4-s + (0.837 + 0.545i)5-s + (0.378 − 0.785i)6-s + (0.378 − 0.925i)7-s + (−0.333 − 0.116i)8-s + (0.482 − 0.189i)9-s + (0.662 − 0.247i)10-s + (−0.175 + 0.446i)11-s + (−0.287 − 0.544i)12-s + (−0.467 − 0.0527i)13-s + (−0.367 − 0.604i)14-s + (1.13 + 0.469i)15-s + (−0.206 + 0.140i)16-s + (−0.112 − 0.00419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.42452 - 1.31602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42452 - 1.31602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.593 + 0.804i)T \) |
| 5 | \( 1 + (-1.87 - 1.22i)T \) |
| 7 | \( 1 + (-1.00 + 2.44i)T \) |
| good | 3 | \( 1 + (-2.09 + 0.396i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (0.581 - 1.48i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (1.68 + 0.190i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (0.461 + 0.0172i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (2.81 - 4.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.160 + 4.27i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-2.65 + 0.606i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.328 + 0.189i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.27 - 0.672i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-3.70 - 7.69i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (3.23 + 9.25i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (1.81 + 1.33i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-1.94 - 1.02i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.226 + 3.01i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (1.74 - 5.65i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (8.24 + 2.20i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.994 - 4.35i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.515 + 0.380i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-9.78 - 5.64i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.87 + 16.6i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (2.73 + 6.96i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 11.1i)T + 97iT^{2} \) |
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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
−10.45929751345145393296187568642, −10.21890870296099643223222634679, −9.149618222461607389120407436876, −8.148679249290998610847132583362, −7.24555737155694905607576806064, −6.21767369475112836771220442875, −4.85498867454120673542266480853, −3.73096369190487288732850451115, −2.62181735309362750005398768692, −1.74423804495072416912947320745,
2.14804311199177180038731787627, 3.05011193831105289451746427328, 4.52231450914302583665384845606, 5.40342478963664970733940996012, 6.34897052921070065075534009377, 7.67282705433339168060690347958, 8.595138509992267652505000255893, 9.015289481390782522342451418785, 9.790133973358380162721980722343, 11.15646731732836516486503488643
