L(s) = 1 | + 3.07·3-s + (1.47 − 1.47i)7-s + 6.45·9-s + (1.20 + 1.20i)11-s + 5.63i·13-s + (4.22 − 4.22i)17-s + (−3.11 − 3.11i)19-s + (4.54 − 4.54i)21-s + (−1.08 − 1.08i)23-s + 10.6·27-s + (−5.32 + 5.32i)29-s − 4.67i·31-s + (3.71 + 3.71i)33-s + 1.51i·37-s + 17.3i·39-s + ⋯ |
L(s) = 1 | + 1.77·3-s + (0.558 − 0.558i)7-s + 2.15·9-s + (0.363 + 0.363i)11-s + 1.56i·13-s + (1.02 − 1.02i)17-s + (−0.715 − 0.715i)19-s + (0.991 − 0.991i)21-s + (−0.225 − 0.225i)23-s + 2.04·27-s + (−0.988 + 0.988i)29-s − 0.839i·31-s + (0.646 + 0.646i)33-s + 0.248i·37-s + 2.77i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.557554008\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.557554008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 7 | \( 1 + (-1.47 + 1.47i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.20 - 1.20i)T + 11iT^{2} \) |
| 13 | \( 1 - 5.63iT - 13T^{2} \) |
| 17 | \( 1 + (-4.22 + 4.22i)T - 17iT^{2} \) |
| 19 | \( 1 + (3.11 + 3.11i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.08 + 1.08i)T + 23iT^{2} \) |
| 29 | \( 1 + (5.32 - 5.32i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.67iT - 31T^{2} \) |
| 37 | \( 1 - 1.51iT - 37T^{2} \) |
| 41 | \( 1 - 3.19iT - 41T^{2} \) |
| 43 | \( 1 - 2.42iT - 43T^{2} \) |
| 47 | \( 1 + (0.827 + 0.827i)T + 47iT^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 + (-7.78 + 7.78i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.03 + 3.03i)T + 61iT^{2} \) |
| 67 | \( 1 + 2.93iT - 67T^{2} \) |
| 71 | \( 1 - 0.180T + 71T^{2} \) |
| 73 | \( 1 + (2.19 - 2.19i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 8.33T + 83T^{2} \) |
| 89 | \( 1 - 9.08T + 89T^{2} \) |
| 97 | \( 1 + (6.04 - 6.04i)T - 97iT^{2} \) |
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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
−9.449656274908462258445861215895, −8.652288994655516374052104732339, −7.83326423911230229384616057307, −7.25260481740936516357846242872, −6.54534254263020026703170508281, −4.87149850434408750537857765102, −4.22179230025744346892864929053, −3.38902396858259835291833177574, −2.29872520376002400082366996531, −1.46715554648824207600370593963,
1.45739631010350081514984668945, 2.40166820721442386536169150649, 3.41718304585976550564715731008, 3.94783405640365848019346141181, 5.31962699478469174226972904952, 6.12447570165414064860163992239, 7.48393586833972277831547071743, 8.037560943210935270771180100679, 8.462844909588221856882642505676, 9.194893611938556231341861625535