L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − i·8-s − 9-s − 4·11-s + i·12-s − 6i·13-s + 16-s + 2i·17-s − i·18-s − 4·19-s − 4i·22-s + 8i·23-s − 24-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s + 0.288i·12-s − 1.66i·13-s + 0.250·16-s + 0.485i·17-s − 0.235i·18-s − 0.917·19-s − 0.852i·22-s + 1.66i·23-s − 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.112846530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.112846530\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−8.131172091433443907611229110211, −7.64225774108516106770381217401, −7.28985638622406679814529895354, −6.03265036783062671656038008801, −5.73308360394574246755242445904, −5.05503011365676459693430618962, −3.93407613333581363368019575718, −3.06084798008186798510762988705, −2.10672157202498530092111965838, −0.75200811348567704805352024882,
0.42491837607475486422221526359, 2.10071159374473486521558737460, 2.51835469249194975979516482251, 3.72116992769178786307264091475, 4.36700328846565953683990333043, 4.97036767889182386816932168076, 5.84179288751046260251331246618, 6.73066408545719865337403776170, 7.52881854441892907560908322998, 8.525647481673992696530012265165