L(s) = 1 | + (0.167 + 0.167i)2-s + (0.707 + 0.707i)3-s − 1.94i·4-s + 0.236i·6-s + (0.0627 + 2.64i)7-s + (0.658 − 0.658i)8-s + 1.00i·9-s + 3.98·11-s + (1.37 − 1.37i)12-s + (0.500 + 0.500i)13-s + (−0.431 + 0.452i)14-s − 3.66·16-s + (−1.67 + 1.67i)17-s + (−0.167 + 0.167i)18-s + 7.21·19-s + ⋯ |
L(s) = 1 | + (0.118 + 0.118i)2-s + (0.408 + 0.408i)3-s − 0.972i·4-s + 0.0964i·6-s + (0.0237 + 0.999i)7-s + (0.232 − 0.232i)8-s + 0.333i·9-s + 1.20·11-s + (0.396 − 0.396i)12-s + (0.138 + 0.138i)13-s + (−0.115 + 0.120i)14-s − 0.917·16-s + (−0.407 + 0.407i)17-s + (−0.0393 + 0.0393i)18-s + 1.65·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86923 + 0.240131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86923 + 0.240131i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.0627 - 2.64i)T \) |
good | 2 | \( 1 + (-0.167 - 0.167i)T + 2iT^{2} \) |
| 11 | \( 1 - 3.98T + 11T^{2} \) |
| 13 | \( 1 + (-0.500 - 0.500i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.67 - 1.67i)T - 17iT^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 + (-5.16 + 5.16i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 - 4.93iT - 31T^{2} \) |
| 37 | \( 1 + (0.292 + 0.292i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.63iT - 41T^{2} \) |
| 43 | \( 1 + (3.65 - 3.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.305 - 0.305i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.39 - 5.39i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 - 7.11iT - 61T^{2} \) |
| 67 | \( 1 + (0.944 + 0.944i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + (-1.38 - 1.38i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.64iT - 79T^{2} \) |
| 83 | \( 1 + (11.9 + 11.9i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 + (-7.43 + 7.43i)T - 97iT^{2} \) |
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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
−10.87508678106785940374935654669, −9.828528760773129533212350426182, −9.168436506232664929318320492965, −8.573952846970246456565201277805, −7.09659900256122732185745173777, −6.19593834325703477339399965111, −5.28232865616807531736055965753, −4.33063957838689459831749959797, −2.93115395716072891321379712276, −1.49109360051312326490333397566,
1.34855093321942638468764334372, 3.10577318938503888844455859723, 3.77864402965104487677564410719, 4.95997484460317782175153826673, 6.61474350320082540964741198712, 7.28448728136676736440392404738, 7.959473958907590513651388044189, 9.076592343278929585738829195322, 9.700614381267880972299920535754, 11.21344994965257865154651060580