L(s) = 1 | + (1.69 + 1.69i)2-s + (−0.366 − 1.69i)3-s + 3.73i·4-s + (1.69 + 1.69i)5-s + (2.24 − 3.48i)6-s + (1 + i)7-s + (−2.93 + 2.93i)8-s + (−2.73 + 1.23i)9-s + 5.73i·10-s + (1.23 − 1.23i)11-s + (6.31 − 1.36i)12-s + 3.38i·14-s + (2.24 − 3.48i)15-s − 2.46·16-s − 2.14·17-s + (−6.72 − 2.52i)18-s + ⋯ |
L(s) = 1 | + (1.19 + 1.19i)2-s + (−0.211 − 0.977i)3-s + 1.86i·4-s + (0.757 + 0.757i)5-s + (0.917 − 1.42i)6-s + (0.377 + 0.377i)7-s + (−1.03 + 1.03i)8-s + (−0.910 + 0.413i)9-s + 1.81i·10-s + (0.373 − 0.373i)11-s + (1.82 − 0.394i)12-s + 0.904i·14-s + (0.580 − 0.899i)15-s − 0.616·16-s − 0.520·17-s + (−1.58 − 0.595i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0809 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.04281 + 1.88357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.04281 + 1.88357i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.366 + 1.69i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.69 - 1.69i)T + 2iT^{2} \) |
| 5 | \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \) |
| 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.23 + 1.23i)T - 11iT^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 + (0.732 - 0.732i)T - 19iT^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.53iT - 29T^{2} \) |
| 31 | \( 1 + (-4.46 + 4.46i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.83 + 4.83i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.453 + 0.453i)T + 41iT^{2} \) |
| 43 | \( 1 + 8.19iT - 43T^{2} \) |
| 47 | \( 1 + (6.77 - 6.77i)T - 47iT^{2} \) |
| 53 | \( 1 + 4.62iT - 53T^{2} \) |
| 59 | \( 1 + (-3.38 + 3.38i)T - 59iT^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + (-6.19 + 6.19i)T - 67iT^{2} \) |
| 71 | \( 1 + (3.38 + 3.38i)T + 71iT^{2} \) |
| 73 | \( 1 + (6.09 + 6.09i)T + 73iT^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (1.23 + 1.23i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7.10 + 7.10i)T - 89iT^{2} \) |
| 97 | \( 1 + (-9.19 + 9.19i)T - 97iT^{2} \) |
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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
−11.45872541136479552848032836978, −10.44912926369867296962887075926, −8.913647372841270370225569311534, −8.034977275689228911857854014208, −7.06267455031392739395373305675, −6.41952744305079114332801955552, −5.80715997734527567379199314418, −4.88338450305186718858226156639, −3.36559672767332452165088586022, −2.10265355031866332586722121472,
1.42527144426796813949704219879, 2.81146271871794290744135972907, 4.11078235732421747882092760109, 4.71609598732893158752847633732, 5.46178848330872228734119450473, 6.48601875866389231109836527807, 8.368381556001954407189146315008, 9.431138584018305743329955415144, 10.04127506622874608156560958333, 10.82923638207396445857889857738