L(s) = 1 | − 2.69·2-s + i·3-s + 5.27·4-s − 2.58·5-s − 2.69i·6-s + (−0.551 − 2.58i)7-s − 8.81·8-s − 9-s + 6.97·10-s − 3.13i·11-s + 5.27i·12-s + 4.69i·13-s + (1.48 + 6.97i)14-s − 2.58i·15-s + 13.2·16-s + 6.97·17-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 0.577i·3-s + 2.63·4-s − 1.15·5-s − 1.10i·6-s + (−0.208 − 0.978i)7-s − 3.11·8-s − 0.333·9-s + 2.20·10-s − 0.946i·11-s + 1.52i·12-s + 1.30i·13-s + (0.397 + 1.86i)14-s − 0.668i·15-s + 3.30·16-s + 1.69·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.247092 + 0.233084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.247092 + 0.233084i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (0.551 + 2.58i)T \) |
| 23 | \( 1 + (1.27 - 4.62i)T \) |
good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 11 | \( 1 + 3.13iT - 11T^{2} \) |
| 13 | \( 1 - 4.69iT - 13T^{2} \) |
| 17 | \( 1 - 6.97T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 + 2.57iT - 31T^{2} \) |
| 37 | \( 1 - 2.90iT - 37T^{2} \) |
| 41 | \( 1 - 5iT - 41T^{2} \) |
| 43 | \( 1 - 0.785iT - 43T^{2} \) |
| 47 | \( 1 - 12.3iT - 47T^{2} \) |
| 53 | \( 1 - 7.52iT - 53T^{2} \) |
| 59 | \( 1 + 3.15iT - 59T^{2} \) |
| 61 | \( 1 - 8.22T + 61T^{2} \) |
| 67 | \( 1 - 7.76iT - 67T^{2} \) |
| 71 | \( 1 - 9.23T + 71T^{2} \) |
| 73 | \( 1 - 9.65iT - 73T^{2} \) |
| 79 | \( 1 - 1.50iT - 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 7.74T + 97T^{2} \) |
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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
−11.16099900487565929821542171023, −10.06907832027414684039160330542, −9.543112409144089584741048907841, −8.429369146890732584577067356708, −7.88480468181792025482846787873, −7.06719979451556161976227849716, −6.03691231366831957335531630565, −4.11263280042823089515400698507, −3.11942131376799219835572476234, −1.09091186066105895264905118689,
0.46040183125985371346753871338, 2.14013534385528242164906253730, 3.28964389272066275679358297939, 5.49578084875635693962682932511, 6.64746784152234966053154502564, 7.51066975347638164129394991671, 8.180456897714614610164673000444, 8.662979868328481918081262225839, 9.912102722962730775476656588870, 10.47165332878437366895139007150