L(s) = 1 | + (0.707 − 0.707i)3-s + (1.95 − 1.09i)5-s + 4.81i·7-s − 1.00i·9-s + (1.53 + 1.53i)11-s + (−0.151 + 3.60i)13-s + (0.606 − 2.15i)15-s + (−3.32 + 3.32i)17-s + (4.77 + 4.77i)19-s + (3.40 + 3.40i)21-s + (−3.58 − 3.58i)23-s + (2.60 − 4.26i)25-s + (−0.707 − 0.707i)27-s − 3.23i·29-s + (−2.30 + 2.30i)31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.872 − 0.488i)5-s + 1.81i·7-s − 0.333i·9-s + (0.462 + 0.462i)11-s + (−0.0420 + 0.999i)13-s + (0.156 − 0.555i)15-s + (−0.805 + 0.805i)17-s + (1.09 + 1.09i)19-s + (0.742 + 0.742i)21-s + (−0.747 − 0.747i)23-s + (0.521 − 0.853i)25-s + (−0.136 − 0.136i)27-s − 0.599i·29-s + (−0.414 + 0.414i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95277 + 0.531052i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95277 + 0.531052i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.95 + 1.09i)T \) |
| 13 | \( 1 + (0.151 - 3.60i)T \) |
good | 7 | \( 1 - 4.81iT - 7T^{2} \) |
| 11 | \( 1 + (-1.53 - 1.53i)T + 11iT^{2} \) |
| 17 | \( 1 + (3.32 - 3.32i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.77 - 4.77i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.58 + 3.58i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.23iT - 29T^{2} \) |
| 31 | \( 1 + (2.30 - 2.30i)T - 31iT^{2} \) |
| 37 | \( 1 + 6.36iT - 37T^{2} \) |
| 41 | \( 1 + (-2.79 + 2.79i)T - 41iT^{2} \) |
| 43 | \( 1 + (0.728 + 0.728i)T + 43iT^{2} \) |
| 47 | \( 1 - 1.10iT - 47T^{2} \) |
| 53 | \( 1 + (-7.40 + 7.40i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.82 + 9.82i)T - 59iT^{2} \) |
| 61 | \( 1 + 0.306T + 61T^{2} \) |
| 67 | \( 1 - 1.00T + 67T^{2} \) |
| 71 | \( 1 + (-9.17 + 9.17i)T - 71iT^{2} \) |
| 73 | \( 1 + 7.22T + 73T^{2} \) |
| 79 | \( 1 + 13.9iT - 79T^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 + (4.25 - 4.25i)T - 89iT^{2} \) |
| 97 | \( 1 + 2.87T + 97T^{2} \) |
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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
−10.05862967781619505974440177305, −9.305865018752600910550949868794, −8.828815720626197902416434260822, −8.058421750885413991610717413194, −6.67854064051459560002780529998, −6.01095589242991968622728425195, −5.19969348160787164523292128125, −3.91103098913983913915986668555, −2.28279420145391133702703153441, −1.84212214183905409565720040642,
1.04919398778249584150842594574, 2.76802522743145376042587022012, 3.64520351271384253364818878065, 4.73058468957940180263689798781, 5.77627586839335227351271104514, 7.01983844555511364964582690167, 7.39501071873260937601591254452, 8.636510305796093228807984437121, 9.668085995409543646542168096092, 10.06969363770017768363660789100