L(s) = 1 | + 1.73i·7-s + 7·13-s + 8.66i·19-s + 5·25-s − 10.3i·31-s − 37-s + 10.3i·43-s + 4·49-s − 13·61-s − 12.1i·67-s − 17·73-s − 12.1i·79-s + 12.1i·91-s − 5·97-s − 19.0i·103-s + ⋯ |
L(s) = 1 | + 0.654i·7-s + 1.94·13-s + 1.98i·19-s + 25-s − 1.86i·31-s − 0.164·37-s + 1.58i·43-s + 0.571·49-s − 1.66·61-s − 1.48i·67-s − 1.98·73-s − 1.36i·79-s + 1.27i·91-s − 0.507·97-s − 1.87i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41476 + 0.379085i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41476 + 0.379085i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8.66iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 17T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−11.21201751875047913058671935927, −10.41623439274679055973854166688, −9.338906886207236034516955166648, −8.490842121873430371566472986357, −7.74239137126834312935054998100, −6.21899907388623648826181652843, −5.80810051314631591306814952085, −4.27641933862465300523732268190, −3.19359405981306094723759491957, −1.55995293248542508477899289659,
1.13293753812297903734402838270, 3.01760839698715801962750927644, 4.14149037091446659297479417917, 5.27988116682589768936923727974, 6.55213855881737620069911805785, 7.20217671530449479560334293127, 8.599536346141706453549999928472, 9.005275133993387819880779534966, 10.52233958930717249313894169368, 10.85264818012528431013622729626