L(s) = 1 | + 3-s − 5-s + (1.18 − 2.05i)7-s + 9-s + (2.18 − 3.78i)11-s + (0.5 + 0.866i)13-s − 15-s + (−0.813 − 1.40i)17-s + (−3.18 − 5.51i)19-s + (1.18 − 2.05i)21-s + (−0.813 − 1.40i)23-s + 25-s + 27-s + (−2.18 + 3.78i)29-s + (−3.87 + 6.70i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + (0.448 − 0.776i)7-s + 0.333·9-s + (0.659 − 1.14i)11-s + (0.138 + 0.240i)13-s − 0.258·15-s + (−0.197 − 0.341i)17-s + (−0.730 − 1.26i)19-s + (0.258 − 0.448i)21-s + (−0.169 − 0.293i)23-s + 0.200·25-s + 0.192·27-s + (−0.405 + 0.703i)29-s + (−0.695 + 1.20i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.714086349\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.714086349\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (1.05 - 8.11i)T \) |
good | 7 | \( 1 + (-1.18 + 2.05i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.18 + 3.78i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.813 + 1.40i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.18 + 5.51i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.813 + 1.40i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.18 - 3.78i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.87 - 6.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.55 + 7.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.18 + 8.98i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 3.37T + 43T^{2} \) |
| 47 | \( 1 + (-0.813 + 1.40i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 8.74T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (6.55 - 11.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.12 + 3.68i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.55 + 6.16i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.384745820018553517931798887991, −7.24089641062311274471567813446, −7.11313645508625652654770102796, −6.06038684939817165593009062720, −5.06091092285716598876781529798, −4.19053943601754495843404879473, −3.66618168883153304172944837339, −2.73890376549569602520223501379, −1.56953958991253918438757652580, −0.44014533643502251072354119628,
1.55739000156374333286696310356, 2.17431635693623432430883163499, 3.31116731657755185330221002373, 4.16338147410362468567307596687, 4.70735929491790800629947870084, 5.86619216306706806704030333286, 6.42059135112005379161788242397, 7.54561703030185104128230336467, 7.904036712225606953059634104335, 8.623851269664806660552202448634