L(s) = 1 | + (−0.965 + 0.258i)2-s + (−1.38 − 1.04i)3-s + (0.866 − 0.499i)4-s + (−1.68 − 1.68i)5-s + (1.60 + 0.653i)6-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (0.809 + 2.88i)9-s + (2.06 + 1.19i)10-s + (1.39 + 5.21i)11-s + (−1.71 − 0.216i)12-s + (−3.48 − 0.938i)13-s − i·14-s + (0.561 + 4.08i)15-s + (0.500 − 0.866i)16-s + (−0.157 − 0.272i)17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (−0.796 − 0.604i)3-s + (0.433 − 0.249i)4-s + (−0.753 − 0.753i)5-s + (0.654 + 0.266i)6-s + (−0.0978 + 0.365i)7-s + (−0.249 + 0.249i)8-s + (0.269 + 0.962i)9-s + (0.652 + 0.376i)10-s + (0.421 + 1.57i)11-s + (−0.496 − 0.0624i)12-s + (−0.965 − 0.260i)13-s − 0.267i·14-s + (0.145 + 1.05i)15-s + (0.125 − 0.216i)16-s + (−0.0381 − 0.0660i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.621788 + 0.0299663i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.621788 + 0.0299663i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 + (1.38 + 1.04i)T \) |
| 7 | \( 1 + (0.258 - 0.965i)T \) |
| 13 | \( 1 + (3.48 + 0.938i)T \) |
good | 5 | \( 1 + (1.68 + 1.68i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.39 - 5.21i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (0.157 + 0.272i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.03 - 0.812i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.97 + 6.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-8.99 - 5.19i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.93 - 1.93i)T - 31iT^{2} \) |
| 37 | \( 1 + (-5.26 + 1.41i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (7.15 - 1.91i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.91 + 2.26i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.31 + 1.31i)T - 47iT^{2} \) |
| 53 | \( 1 - 13.4iT - 53T^{2} \) |
| 59 | \( 1 + (-10.6 - 2.84i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (1.00 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.360 - 1.34i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.98 + 14.8i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-10.3 - 10.3i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.912T + 79T^{2} \) |
| 83 | \( 1 + (-7.71 - 7.71i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.444 + 1.65i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-10.5 - 2.83i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.74351234944312052840769290336, −9.959727018322508506313938701562, −8.968756764347930618950058891801, −8.038524404629149319920378986372, −7.21502553085267574447667081439, −6.58355237070186487546391529888, −5.14568398505521371484963503699, −4.56782244477120598399215089839, −2.44162237810997935161330487425, −0.941866038747187988611510900912,
0.70394064708994464610304667356, 3.07153800563458057892503770541, 3.84456884660509909730559904291, 5.22989416788712365868846186001, 6.41373398476706721837814192361, 7.15097905621092894945346753458, 8.121413134158885674898573762563, 9.278488417433704406376672898696, 9.976810148997995037817618136624, 10.88236430339733484135416114013