L(s) = 1 | + 0.585·3-s − 5-s − 2.65·9-s − 4.82·11-s − 0.828·13-s − 0.585·15-s − 5.41·17-s + 3.41·19-s + 6.82·23-s + 25-s − 3.31·27-s + 0.828·29-s + 2.82·31-s − 2.82·33-s + 3.65·37-s − 0.485·39-s − 11.0·41-s + 3.17·43-s + 2.65·45-s + 10.8·47-s − 3.17·51-s − 10.4·53-s + 4.82·55-s + 2·57-s + 11.4·59-s + 13.3·61-s + 0.828·65-s + ⋯ |
L(s) = 1 | + 0.338·3-s − 0.447·5-s − 0.885·9-s − 1.45·11-s − 0.229·13-s − 0.151·15-s − 1.31·17-s + 0.783·19-s + 1.42·23-s + 0.200·25-s − 0.637·27-s + 0.153·29-s + 0.508·31-s − 0.492·33-s + 0.601·37-s − 0.0777·39-s − 1.72·41-s + 0.483·43-s + 0.396·45-s + 1.57·47-s − 0.444·51-s − 1.44·53-s + 0.651·55-s + 0.264·57-s + 1.48·59-s + 1.70·61-s + 0.102·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.269176941\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.269176941\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.585T + 3T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 6.82T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 3.65T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 3.17T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 - 1.17T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.498721180093561669510885365990, −7.78553553643322768018561132562, −7.15178722084264799934689833095, −6.31317563288933886098026474916, −5.24575071850530207553389623313, −4.90164310537265708096105500017, −3.72188982284049364514294833987, −2.85082410284428121753506914211, −2.30387834697612667523824269756, −0.60511987838352224882598091851,
0.60511987838352224882598091851, 2.30387834697612667523824269756, 2.85082410284428121753506914211, 3.72188982284049364514294833987, 4.90164310537265708096105500017, 5.24575071850530207553389623313, 6.31317563288933886098026474916, 7.15178722084264799934689833095, 7.78553553643322768018561132562, 8.498721180093561669510885365990