L(s) = 1 | − 2-s + (−1.64 + 0.543i)3-s + 4-s + 0.118i·5-s + (1.64 − 0.543i)6-s − i·7-s − 8-s + (2.40 − 1.78i)9-s − 0.118i·10-s + (−2.10 + 2.56i)11-s + (−1.64 + 0.543i)12-s − 0.913i·13-s + i·14-s + (−0.0644 − 0.195i)15-s + 16-s + 0.404·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.949 + 0.313i)3-s + 0.5·4-s + 0.0530i·5-s + (0.671 − 0.221i)6-s − 0.377i·7-s − 0.353·8-s + (0.803 − 0.595i)9-s − 0.0375i·10-s + (−0.633 + 0.773i)11-s + (−0.474 + 0.156i)12-s − 0.253i·13-s + 0.267i·14-s + (−0.0166 − 0.0503i)15-s + 0.250·16-s + 0.0980·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.682800 - 0.126700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.682800 - 0.126700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.64 - 0.543i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (2.10 - 2.56i)T \) |
good | 5 | \( 1 - 0.118iT - 5T^{2} \) |
| 13 | \( 1 + 0.913iT - 13T^{2} \) |
| 17 | \( 1 - 0.404T + 17T^{2} \) |
| 19 | \( 1 + 7.70iT - 19T^{2} \) |
| 23 | \( 1 - 1.37iT - 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 - 4.13T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 2.85iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 + 8.81iT - 53T^{2} \) |
| 59 | \( 1 + 5.49iT - 59T^{2} \) |
| 61 | \( 1 + 7.30iT - 61T^{2} \) |
| 67 | \( 1 + 2.01T + 67T^{2} \) |
| 71 | \( 1 - 14.2iT - 71T^{2} \) |
| 73 | \( 1 - 11.7iT - 73T^{2} \) |
| 79 | \( 1 - 4.34iT - 79T^{2} \) |
| 83 | \( 1 + 0.700T + 83T^{2} \) |
| 89 | \( 1 + 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 5.48T + 97T^{2} \) |
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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
−10.93584538307928936129976239847, −10.10836014147285385784741413852, −9.500708052349717070989266575327, −8.295293253264523905257784642085, −7.17622095580728330643671138458, −6.58255346915222103252242878011, −5.27080865803373091118369500042, −4.40490549333207608754600492102, −2.70116933774036440153103301331, −0.77481443916706013500227676023,
1.10175870564906438994396887527, 2.72249959726419689929701158461, 4.48813064059235738427450274601, 5.80105272709546164791861896376, 6.29726787722676497973647715974, 7.59730193876409250723583326229, 8.247246558050069027163819851736, 9.380548107476058034363579741099, 10.45031959963739498711077255128, 10.87119969716441703090834262292