L(s) = 1 | − 1.42i·2-s + 3-s − 0.0397·4-s − 0.0606i·5-s − 1.42i·6-s − 1.70i·7-s − 2.79i·8-s + 9-s − 0.0866·10-s + i·11-s − 0.0397·12-s + (3.52 + 0.738i)13-s − 2.43·14-s − 0.0606i·15-s − 4.07·16-s − 3.75·17-s + ⋯ |
L(s) = 1 | − 1.00i·2-s + 0.577·3-s − 0.0198·4-s − 0.0271i·5-s − 0.583i·6-s − 0.644i·7-s − 0.989i·8-s + 0.333·9-s − 0.0273·10-s + 0.301i·11-s − 0.0114·12-s + (0.978 + 0.204i)13-s − 0.651·14-s − 0.0156i·15-s − 1.01·16-s − 0.910·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17348 - 1.44444i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17348 - 1.44444i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-3.52 - 0.738i)T \) |
good | 2 | \( 1 + 1.42iT - 2T^{2} \) |
| 5 | \( 1 + 0.0606iT - 5T^{2} \) |
| 7 | \( 1 + 1.70iT - 7T^{2} \) |
| 17 | \( 1 + 3.75T + 17T^{2} \) |
| 19 | \( 1 + 2.02iT - 19T^{2} \) |
| 23 | \( 1 + 0.704T + 23T^{2} \) |
| 29 | \( 1 + 0.0346T + 29T^{2} \) |
| 31 | \( 1 + 1.85iT - 31T^{2} \) |
| 37 | \( 1 - 8.82iT - 37T^{2} \) |
| 41 | \( 1 + 3.32iT - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 - 6.04iT - 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 3.34iT - 59T^{2} \) |
| 61 | \( 1 - 3.26T + 61T^{2} \) |
| 67 | \( 1 - 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 1.96iT - 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 4.19iT - 83T^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 + 5.86iT - 97T^{2} \) |
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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
−10.93238376607560255482925802290, −10.18467038853085726941499780358, −9.280927799198286336048733680358, −8.381457916197158578566280326651, −7.14210775191071345324748184735, −6.43491688752738045578483380283, −4.60536327221685497106649791888, −3.69133482550734492073584201251, −2.59497172826488486684952458292, −1.28293401292494700422053799570,
2.06132777468751451878608717052, 3.37923803072449927445123238955, 4.88988733306491757804696045746, 5.98089640186134828577182169959, 6.69937063470640473028116709772, 7.79528606982338850504969311243, 8.573558917508861084022221161173, 9.138511178296992250131426622046, 10.58380355004422770085359126018, 11.31103253308247141360558306092