L(s) = 1 | + (0.360 + 1.69i)3-s + (−1.61 + 1.17i)4-s + (1.88 + 0.837i)7-s + (−2.74 + 1.22i)9-s + (−2.57 − 2.31i)12-s + (5.30 − 4.77i)13-s + (1.23 − 3.80i)16-s + (−1.74 + 1.93i)19-s + (−0.741 + 3.48i)21-s + (−2.5 − 4.33i)25-s + (−3.05 − 4.20i)27-s + (−4.02 + 0.855i)28-s + (5.55 − 0.296i)31-s + (3.00 − 5.19i)36-s + (−10.4 + 6.03i)37-s + ⋯ |
L(s) = 1 | + (0.207 + 0.978i)3-s + (−0.809 + 0.587i)4-s + (0.710 + 0.316i)7-s + (−0.913 + 0.406i)9-s + (−0.743 − 0.669i)12-s + (1.47 − 1.32i)13-s + (0.309 − 0.951i)16-s + (−0.400 + 0.444i)19-s + (−0.161 + 0.761i)21-s + (−0.5 − 0.866i)25-s + (−0.587 − 0.809i)27-s + (−0.761 + 0.161i)28-s + (0.998 − 0.0532i)31-s + (0.500 − 0.866i)36-s + (−1.71 + 0.991i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.767136 + 0.556053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.767136 + 0.556053i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.360 - 1.69i)T \) |
| 31 | \( 1 + (-5.55 + 0.296i)T \) |
good | 2 | \( 1 + (1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.88 - 0.837i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (-5.30 + 4.77i)T + (1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (1.74 - 1.93i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (10.4 - 6.03i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-6.31 - 5.68i)T + (4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 2.62iT - 61T^{2} \) |
| 67 | \( 1 + (-7.49 + 12.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (15.5 - 1.63i)T + (71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (11.5 + 1.21i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.78 - 4.20i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.22393300214343963544804096237, −13.38993659570027584139756546937, −12.12295591183965126138618775269, −10.90507135998168467913405538257, −9.899129624581495638820549813985, −8.481962763040090094617800351645, −8.197410635356018385709191441887, −5.76574762956726733518220062836, −4.53150263395175964780550367247, −3.28446896734065664730281152784,
1.52666541951277348169336733726, 4.08730349124566849199642013513, 5.70979325045132626554403560203, 6.95902906108432179939819737123, 8.417789438325054974218084501880, 9.115176909062379307616114058469, 10.75965427661621037297515438020, 11.70201205416395250082268342010, 13.10419794021573190048235161162, 13.85458818721137333825887100264