| L(s) = 1 | + (0.387 + 0.934i)2-s + (−2.72 − 1.82i)3-s + (0.690 − 0.690i)4-s + (−2.09 + 3.14i)5-s + (0.647 − 3.25i)6-s + (−0.0152 − 0.0228i)7-s + (2.78 + 1.15i)8-s + (2.97 + 7.18i)9-s + (−3.74 − 0.745i)10-s + (−0.335 + 1.68i)11-s + (−3.14 + 0.625i)12-s + (−2.06 + 2.95i)13-s + (0.0154 − 0.0230i)14-s + (11.4 − 4.74i)15-s + 1.09i·16-s + (−3.51 + 2.15i)17-s + ⋯ |
| L(s) = 1 | + (0.273 + 0.660i)2-s + (−1.57 − 1.05i)3-s + (0.345 − 0.345i)4-s + (−0.938 + 1.40i)5-s + (0.264 − 1.32i)6-s + (−0.00576 − 0.00862i)7-s + (0.983 + 0.407i)8-s + (0.992 + 2.39i)9-s + (−1.18 − 0.235i)10-s + (−0.101 + 0.508i)11-s + (−0.908 + 0.180i)12-s + (−0.572 + 0.819i)13-s + (0.00412 − 0.00616i)14-s + (2.95 − 1.22i)15-s + 0.272i·16-s + (−0.852 + 0.523i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.352086 + 0.523598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.352086 + 0.523598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.06 - 2.95i)T \) |
| 17 | \( 1 + (3.51 - 2.15i)T \) |
| good | 2 | \( 1 + (-0.387 - 0.934i)T + (-1.41 + 1.41i)T^{2} \) |
| 3 | \( 1 + (2.72 + 1.82i)T + (1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (2.09 - 3.14i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (0.0152 + 0.0228i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (0.335 - 1.68i)T + (-10.1 - 4.20i)T^{2} \) |
| 19 | \( 1 + (-0.960 - 2.31i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.920 - 1.37i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.615 - 3.09i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (0.374 + 1.88i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (1.18 + 5.98i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.32 + 0.882i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (6.47 - 2.68i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (3.58 - 8.66i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-8.96 - 3.71i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (2.18 - 10.9i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (4.70 + 4.70i)T + 67iT^{2} \) |
| 71 | \( 1 + (-2.19 + 0.437i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-7.50 - 5.01i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-6.23 - 9.32i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-5.25 + 2.17i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + 10.7iT - 89T^{2} \) |
| 97 | \( 1 + (-10.3 - 6.92i)T + (37.1 + 89.6i)T^{2} \) |
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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
−12.31691309448417198519812360540, −11.54804107207947262023111011868, −10.95550451489939665506756542749, −10.23695372064362446120497493465, −7.82748484381198673854155486690, −7.04589460391502574294012621398, −6.73131170851178770635618826389, −5.69750412270747013693899169779, −4.44261631582941064995756006089, −2.04044035104993676048496806505,
0.57731808476894690216319370017, 3.49810109263079967402855685178, 4.64320757683882674219906527100, 5.10897640409196966851043173904, 6.67979592259964548140272794193, 8.069003105645082111674855920602, 9.332210591599359075553599659480, 10.42091744037333459697512174824, 11.31925599475388100268016820046, 11.78463634037341764251988329351
