L(s) = 1 | + 2.75·3-s + (−4.30 − 2.54i)5-s − 3.84·7-s − 1.41·9-s − 6.19i·11-s − 16.1i·13-s + (−11.8 − 7.01i)15-s + 5.20i·17-s − 36.2i·19-s − 10.6·21-s − 22.0·23-s + (12.0 + 21.9i)25-s − 28.6·27-s + 20.0·29-s − 26.4i·31-s + ⋯ |
L(s) = 1 | + 0.918·3-s + (−0.860 − 0.509i)5-s − 0.549·7-s − 0.157·9-s − 0.562i·11-s − 1.23i·13-s + (−0.789 − 0.467i)15-s + 0.306i·17-s − 1.90i·19-s − 0.504·21-s − 0.958·23-s + (0.480 + 0.876i)25-s − 1.06·27-s + 0.690·29-s − 0.852i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.569309 - 0.998800i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.569309 - 0.998800i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (4.30 + 2.54i)T \) |
good | 3 | \( 1 - 2.75T + 9T^{2} \) |
| 7 | \( 1 + 3.84T + 49T^{2} \) |
| 11 | \( 1 + 6.19iT - 121T^{2} \) |
| 13 | \( 1 + 16.1iT - 169T^{2} \) |
| 17 | \( 1 - 5.20iT - 289T^{2} \) |
| 19 | \( 1 + 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 22.0T + 529T^{2} \) |
| 29 | \( 1 - 20.0T + 841T^{2} \) |
| 31 | \( 1 + 26.4iT - 961T^{2} \) |
| 37 | \( 1 - 69.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 25.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 39.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 27.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 54.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 107.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 37.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 97.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 126.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 6.40iT - 9.40e3T^{2} \) |
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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
−11.19545579304025996830698510031, −10.04948548997752523545772696659, −8.967176434086083931896883545396, −8.330812386426172644270936008568, −7.58447396936173104882702383564, −6.23196446552555725300768410311, −4.90908254807366358396367347867, −3.57897855638254437751457411867, −2.74772341855579465579377837742, −0.46868823325521407782078802965,
2.15288949693950947600483807677, 3.44365412428792927456027975228, 4.22750422705702766620509826304, 6.00523295558657510294123113388, 7.12567616712491269998000023310, 7.927834244479482774302270786527, 8.855197806449527835423536015335, 9.767933290161720482186019582461, 10.72283289179252834994049702905, 11.94597275846186719167373695704