L(s) = 1 | + 2.31i·5-s + (−0.222 − 2.63i)7-s + 3.58i·11-s − 2.82i·13-s − 2.31i·17-s − 7.90·19-s + 0.130i·23-s − 0.355·25-s + 0.199·29-s + 3.45·31-s + (6.10 − 0.514i)35-s + 11.5·37-s − 7.97i·41-s + 4.38i·43-s + 6.54·47-s + ⋯ |
L(s) = 1 | + 1.03i·5-s + (−0.0839 − 0.996i)7-s + 1.08i·11-s − 0.784i·13-s − 0.561i·17-s − 1.81·19-s + 0.0271i·23-s − 0.0711·25-s + 0.0371·29-s + 0.620·31-s + (1.03 − 0.0869i)35-s + 1.90·37-s − 1.24i·41-s + 0.668i·43-s + 0.954·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.699551935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.699551935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.222 + 2.63i)T \) |
good | 5 | \( 1 - 2.31iT - 5T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 2.31iT - 17T^{2} \) |
| 19 | \( 1 + 7.90T + 19T^{2} \) |
| 23 | \( 1 - 0.130iT - 23T^{2} \) |
| 29 | \( 1 - 0.199T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 7.97iT - 41T^{2} \) |
| 43 | \( 1 - 4.38iT - 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 + 6.19iT - 61T^{2} \) |
| 67 | \( 1 - 9.01iT - 67T^{2} \) |
| 71 | \( 1 - 4.75iT - 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 4.24iT - 79T^{2} \) |
| 83 | \( 1 + 0.768T + 83T^{2} \) |
| 89 | \( 1 - 17.7iT - 89T^{2} \) |
| 97 | \( 1 - 9.02iT - 97T^{2} \) |
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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
−8.264033747045428919674039531255, −7.61062840525673215927126045592, −6.92657147924608983272211889901, −6.53151041557113560498557146080, −5.53525975750279732084273701612, −4.44735483199970370504935049590, −3.98153586145183885195897183357, −2.85685289846444022902768906882, −2.19584967488733657070802555864, −0.70363953043353346940643913160,
0.77606582142321662862091261809, 1.97233472988403162082796243801, 2.80631426209119781256646187835, 4.05593646282766007453009187614, 4.57041715123240760797341306139, 5.59675057360595211343764969040, 6.08323330261651669737783341351, 6.78884388708101202782780464769, 8.071579193991006555500179665687, 8.549803150830717640594735676412