L(s) = 1 | + (−0.595 − 0.595i)5-s − 1.64i·7-s + (3.36 + 3.36i)11-s + (−2.64 + 2.64i)13-s + 5.53·17-s + (−3.64 + 3.64i)19-s + 4.33i·23-s − 4.29i·25-s + (6.12 − 6.12i)29-s + 5.64·31-s + (−0.979 + 0.979i)35-s + (0.645 + 0.645i)37-s − 7.91i·41-s + (−0.354 − 0.354i)43-s + 9.10·47-s + ⋯ |
L(s) = 1 | + (−0.266 − 0.266i)5-s − 0.622i·7-s + (1.01 + 1.01i)11-s + (−0.733 + 0.733i)13-s + 1.34·17-s + (−0.836 + 0.836i)19-s + 0.904i·23-s − 0.858i·25-s + (1.13 − 1.13i)29-s + 1.01·31-s + (−0.165 + 0.165i)35-s + (0.106 + 0.106i)37-s − 1.23i·41-s + (−0.0540 − 0.0540i)43-s + 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.615225749\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.615225749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.595 + 0.595i)T + 5iT^{2} \) |
| 7 | \( 1 + 1.64iT - 7T^{2} \) |
| 11 | \( 1 + (-3.36 - 3.36i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.64 - 2.64i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + (3.64 - 3.64i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.33iT - 23T^{2} \) |
| 29 | \( 1 + (-6.12 + 6.12i)T - 29iT^{2} \) |
| 31 | \( 1 - 5.64T + 31T^{2} \) |
| 37 | \( 1 + (-0.645 - 0.645i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.91iT - 41T^{2} \) |
| 43 | \( 1 + (0.354 + 0.354i)T + 43iT^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + (-4.93 - 4.93i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.33 - 4.33i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.645 + 0.645i)T - 61iT^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 - 3.29iT - 73T^{2} \) |
| 79 | \( 1 + 9.64T + 79T^{2} \) |
| 83 | \( 1 + (3.36 - 3.36i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.38iT - 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−10.05714239500210289971466408850, −9.052394898015241711431781612180, −8.106678349821970592949210662412, −7.33355802348935806874298769318, −6.62456812315889779689771141454, −5.58036852560387525591823078088, −4.27891601664286915716661209791, −4.04120942442293586217752641494, −2.39976235403248726819098093211, −1.10838198232179727568753270205,
0.917076945509328655634521290283, 2.64856487520332759208697655957, 3.38838930548990151396166292520, 4.62491484877155547010792680581, 5.58367485063617833802133227917, 6.41712829809162467239032877642, 7.24977444257921264592924641294, 8.323746021104498772477203813778, 8.793032866495406230104173639415, 9.798577376383476103558259085621