L(s) = 1 | + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s + (1.5 − 0.866i)5-s + (1.73 − 2i)7-s + (2 − 1.99i)8-s + (0.633 + 2.36i)10-s + (0.866 + 0.5i)11-s − 3.46i·13-s + (2.09 + 3.09i)14-s + (1.99 + 3.46i)16-s + (1.5 + 0.866i)17-s + (2.59 + 4.5i)19-s − 3.46·20-s + (−1 + 0.999i)22-s + (0.866 − 0.5i)23-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.670 − 0.387i)5-s + (0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + (0.200 + 0.748i)10-s + (0.261 + 0.150i)11-s − 0.960i·13-s + (0.560 + 0.827i)14-s + (0.499 + 0.866i)16-s + (0.363 + 0.210i)17-s + (0.596 + 1.03i)19-s − 0.774·20-s + (−0.213 + 0.213i)22-s + (0.180 − 0.104i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17801 + 0.275962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17801 + 0.275962i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 - 4.5i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-0.866 + 1.5i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + (-4.33 - 7.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14iT - 71T^{2} \) |
| 73 | \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.79 - 4.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + (13.5 - 7.79i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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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
−12.34644501914327901913710598803, −10.83330456132843664996399786308, −10.03063330434424302755149810261, −9.153773890876115466387162209172, −8.001127700191252488468349354212, −7.35453885307732788569458248129, −5.94276809123589883637803203014, −5.20090128908411526690049517406, −3.88970669704056830956576049036, −1.32934057357336482862740897306,
1.74866543200013856322997759698, 2.93246459607072519749466339585, 4.52740840318411717384902245114, 5.65455300907543654002181839811, 7.10986310140454186452835465604, 8.443681402675250788071282963638, 9.258587607965775873298110239317, 10.02133305920299209255437259453, 11.26336100624428210761731770021, 11.65966563460862894801018543287