L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 0.499i)6-s + (−0.758 − 1.31i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−4.45 − 2.57i)11-s + 0.999i·12-s + (−1.86 + 3.08i)13-s + 1.51·14-s + (−0.5 + 0.866i)16-s + (−4.15 + 2.39i)17-s − 0.999·18-s + (1.39 − 0.807i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.353 − 0.204i)6-s + (−0.286 − 0.496i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−1.34 − 0.775i)11-s + 0.288i·12-s + (−0.516 + 0.856i)13-s + 0.405·14-s + (−0.125 + 0.216i)16-s + (−1.00 + 0.581i)17-s − 0.235·18-s + (0.320 − 0.185i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7805028789\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7805028789\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (1.86 - 3.08i)T \) |
good | 7 | \( 1 + (0.758 + 1.31i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.45 + 2.57i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (4.15 - 2.39i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 0.807i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.71 - 2.72i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.93 + 8.55i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.88iT - 31T^{2} \) |
| 37 | \( 1 + (4.36 - 7.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.10 - 0.637i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.04 - 1.18i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6.84T + 47T^{2} \) |
| 53 | \( 1 + 0.881iT - 53T^{2} \) |
| 59 | \( 1 + (-8.04 + 4.64i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.24 + 10.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.76 + 9.98i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.7 + 7.91i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 0.979T + 83T^{2} \) |
| 89 | \( 1 + (-3.54 - 2.04i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.85 - 4.95i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.150704171811977189078861322654, −8.334059144977820391634140840232, −7.67645699872512769601541221472, −6.73873287035463676797661617995, −6.41345615320794306612146585095, −5.20340579482649548376855765979, −4.75709411137341057045669152569, −3.42238329170287864056191686591, −2.15683525603600089974425401368, −0.67837799806466063132188279196,
0.56306300588887861210989723122, 2.34325469406080357286548435977, 2.88031956772704026926351068217, 4.22426481360524093261888289594, 5.08918758500530027913108530545, 5.64232793944466257582722048329, 6.97890954444811112689507906428, 7.49654405203049419445021607952, 8.565853223091701499619422963111, 9.218361511580727723607840899879