L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (1.94 + 1.09i)5-s − 1.00i·6-s + (−1.54 + 1.54i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−2.15 + 0.602i)10-s + 1.86i·11-s + (0.707 + 0.707i)12-s + (−1.01 + 1.01i)13-s − 2.18i·14-s + (−2.15 + 0.602i)15-s − 1.00·16-s + (−4.64 − 4.64i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.871 + 0.490i)5-s − 0.408i·6-s + (−0.583 + 0.583i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (−0.680 + 0.190i)10-s + 0.563i·11-s + (0.204 + 0.204i)12-s + (−0.281 + 0.281i)13-s − 0.583i·14-s + (−0.555 + 0.155i)15-s − 0.250·16-s + (−1.12 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0557534 - 0.615157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0557534 - 0.615157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.94 - 1.09i)T \) |
| 31 | \( 1 + (-5.27 - 1.77i)T \) |
good | 7 | \( 1 + (1.54 - 1.54i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.86iT - 11T^{2} \) |
| 13 | \( 1 + (1.01 - 1.01i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.64 + 4.64i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.65iT - 19T^{2} \) |
| 23 | \( 1 + (2.46 - 2.46i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 37 | \( 1 + (5.38 + 5.38i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 + (3.51 - 3.51i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.19 - 5.19i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.72 - 7.72i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.31iT - 59T^{2} \) |
| 61 | \( 1 + 9.74iT - 61T^{2} \) |
| 67 | \( 1 + (-7.73 + 7.73i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.778T + 71T^{2} \) |
| 73 | \( 1 + (-1.92 + 1.92i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.71T + 79T^{2} \) |
| 83 | \( 1 + (0.710 - 0.710i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.40T + 89T^{2} \) |
| 97 | \( 1 + (3.69 - 3.69i)T - 97iT^{2} \) |
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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
−10.24304542739611185006699557821, −9.557744526376904730568177245337, −9.208563569531201178416605788340, −8.001029179623119467655330534391, −6.79715199852807479502730074624, −6.42898649658104017952507495285, −5.44621722873713302136469578015, −4.62190218867939333153565718523, −3.04771627622104517362893177806, −1.87606489449887525654688466743,
0.34526117524647954448112360933, 1.68630586495341530701126786757, 2.83892092058190882837823570603, 4.20257468705558764005936302506, 5.25699110170823267840524625771, 6.45539478368026949300728616911, 6.81959090572454605576367918991, 8.336772774460639694593352366339, 8.650197936164758062547806980475, 9.966775331521269051296387126934