L(s) = 1 | + 0.805·2-s + (−0.972 + 1.43i)3-s − 1.35·4-s + i·5-s + (−0.783 + 1.15i)6-s + i·7-s − 2.69·8-s + (−1.10 − 2.78i)9-s + 0.805i·10-s + (0.889 − 3.19i)11-s + (1.31 − 1.93i)12-s − 0.669i·13-s + 0.805i·14-s + (−1.43 − 0.972i)15-s + 0.530·16-s − 5.06·17-s + ⋯ |
L(s) = 1 | + 0.569·2-s + (−0.561 + 0.827i)3-s − 0.675·4-s + 0.447i·5-s + (−0.319 + 0.471i)6-s + 0.377i·7-s − 0.954·8-s + (−0.369 − 0.929i)9-s + 0.254i·10-s + (0.268 − 0.963i)11-s + (0.379 − 0.559i)12-s − 0.185i·13-s + 0.215i·14-s + (−0.370 − 0.251i)15-s + 0.132·16-s − 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8565442475\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8565442475\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.972 - 1.43i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.889 + 3.19i)T \) |
good | 2 | \( 1 - 0.805T + 2T^{2} \) |
| 13 | \( 1 + 0.669iT - 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 + 8.50iT - 19T^{2} \) |
| 23 | \( 1 - 6.77iT - 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 1.79iT - 43T^{2} \) |
| 47 | \( 1 + 7.04iT - 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 + 11.6iT - 59T^{2} \) |
| 61 | \( 1 + 7.58iT - 61T^{2} \) |
| 67 | \( 1 + 9.11T + 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 12.3iT - 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 + 6.36T + 83T^{2} \) |
| 89 | \( 1 - 8.93iT - 89T^{2} \) |
| 97 | \( 1 + 9.76T + 97T^{2} \) |
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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
−9.531670035317055599764143012549, −9.075930672025273354069053370951, −8.323171804538863374627104343318, −6.80970645716366094474431275601, −6.12333744971131964307926493860, −5.25024305538391880817691101113, −4.58463386980908445109864254773, −3.60139710980270979321919171013, −2.79591802750606216814509141565, −0.38202455626171568084084902567,
1.20294302346511101863426640694, 2.55249018821733711805943363872, 4.29327112415705442563286993725, 4.49472532951460344383465956196, 5.75204113544588920414142705127, 6.35940169596177854625509607583, 7.33813340664007630387696413838, 8.255760865245816445411898969292, 8.945086137564775047370092062685, 10.00292996632004550864096786632