L(s) = 1 | − 1.88·2-s + (1.68 + 0.400i)3-s + 1.55·4-s + 4.10i·5-s + (−3.17 − 0.754i)6-s + 3.01i·7-s + 0.847·8-s + (2.67 + 1.34i)9-s − 7.73i·10-s + (2.60 − 2.05i)11-s + (2.61 + 0.620i)12-s + i·13-s − 5.68i·14-s + (−1.64 + 6.91i)15-s − 4.69·16-s − 6.56·17-s + ⋯ |
L(s) = 1 | − 1.33·2-s + (0.972 + 0.231i)3-s + 0.775·4-s + 1.83i·5-s + (−1.29 − 0.308i)6-s + 1.14i·7-s + 0.299·8-s + (0.893 + 0.449i)9-s − 2.44i·10-s + (0.783 − 0.620i)11-s + (0.754 + 0.179i)12-s + 0.277i·13-s − 1.52i·14-s + (−0.424 + 1.78i)15-s − 1.17·16-s − 1.59·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.509132 + 0.799257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.509132 + 0.799257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.68 - 0.400i)T \) |
| 11 | \( 1 + (-2.60 + 2.05i)T \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 5 | \( 1 - 4.10iT - 5T^{2} \) |
| 7 | \( 1 - 3.01iT - 7T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 + 1.38iT - 19T^{2} \) |
| 23 | \( 1 + 5.41iT - 23T^{2} \) |
| 29 | \( 1 - 6.07T + 29T^{2} \) |
| 31 | \( 1 - 6.13T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 - 0.966T + 41T^{2} \) |
| 43 | \( 1 - 3.07iT - 43T^{2} \) |
| 47 | \( 1 - 2.62iT - 47T^{2} \) |
| 53 | \( 1 + 7.86iT - 53T^{2} \) |
| 59 | \( 1 - 3.17iT - 59T^{2} \) |
| 61 | \( 1 - 4.19iT - 61T^{2} \) |
| 67 | \( 1 + 0.278T + 67T^{2} \) |
| 71 | \( 1 - 1.72iT - 71T^{2} \) |
| 73 | \( 1 + 3.90iT - 73T^{2} \) |
| 79 | \( 1 + 6.67iT - 79T^{2} \) |
| 83 | \( 1 - 6.62T + 83T^{2} \) |
| 89 | \( 1 - 17.4iT - 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−11.00873271420457977565589491979, −10.41363699751146879222257237804, −9.486919337338246726450037926319, −8.762955201233556085425694144210, −8.164426733279617269463606275295, −6.86878609626161409106971444260, −6.47641829529944540147760193814, −4.35713561851872995470205247049, −2.88746465973678180378597891617, −2.17321651035593169210632378752,
0.892260716845210700782770488981, 1.80223362020164712127647126855, 4.06053655762849293488177366497, 4.68532972271225826814089017875, 6.71346905885122889085535433005, 7.59810907602753771719230534507, 8.387059788143197706459671428223, 8.979057994972080714706209931683, 9.632718982651613926834518641606, 10.38477915471015012910010256734