L(s) = 1 | + (3.98 − 3.98i)3-s + (−2.85 − 2.85i)5-s + (−6.99 − 0.139i)7-s − 22.7i·9-s + (7.34 − 7.34i)11-s + (−7.74 + 7.74i)13-s − 22.7·15-s + 16.9i·17-s + (−9.10 + 9.10i)19-s + (−28.4 + 27.3i)21-s − 38.7i·23-s − 8.66i·25-s + (−54.7 − 54.7i)27-s + (14.6 + 14.6i)29-s − 8.79i·31-s + ⋯ |
L(s) = 1 | + (1.32 − 1.32i)3-s + (−0.571 − 0.571i)5-s + (−0.999 − 0.0199i)7-s − 2.52i·9-s + (0.667 − 0.667i)11-s + (−0.595 + 0.595i)13-s − 1.51·15-s + 0.994i·17-s + (−0.479 + 0.479i)19-s + (−1.35 + 1.30i)21-s − 1.68i·23-s − 0.346i·25-s + (−2.02 − 2.02i)27-s + (0.505 + 0.505i)29-s − 0.283i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.758118823\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.758118823\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (6.99 + 0.139i)T \) |
good | 3 | \( 1 + (-3.98 + 3.98i)T - 9iT^{2} \) |
| 5 | \( 1 + (2.85 + 2.85i)T + 25iT^{2} \) |
| 11 | \( 1 + (-7.34 + 7.34i)T - 121iT^{2} \) |
| 13 | \( 1 + (7.74 - 7.74i)T - 169iT^{2} \) |
| 17 | \( 1 - 16.9iT - 289T^{2} \) |
| 19 | \( 1 + (9.10 - 9.10i)T - 361iT^{2} \) |
| 23 | \( 1 + 38.7iT - 529T^{2} \) |
| 29 | \( 1 + (-14.6 - 14.6i)T + 841iT^{2} \) |
| 31 | \( 1 + 8.79iT - 961T^{2} \) |
| 37 | \( 1 + (5.59 - 5.59i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 0.648T + 1.68e3T^{2} \) |
| 43 | \( 1 + (11.6 - 11.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 5.84iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-34.7 + 34.7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-4.12 - 4.12i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-32.6 + 32.6i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (74.0 + 74.0i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 24.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 95.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 43.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-82.9 + 82.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 93.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 116. iT - 9.40e3T^{2} \) |
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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
−10.29697765673980275085573074309, −9.145418395030859435578971133973, −8.580028679891059493607415264637, −7.924756269686503867282353306167, −6.71348951381484087929354056401, −6.31432665412251095175754863291, −4.20756589441799419927160612448, −3.28903571277979943528056138372, −2.07181830773224175350749220319, −0.60476671055231046144958652970,
2.54473631629186544452235926015, 3.35173570124242324666293289543, 4.14920825653020726933690151923, 5.29087266028491891612799391959, 7.00078944723746775602705627333, 7.65911848724651521886804734132, 8.863544481273258470985584475524, 9.574713149010211962791186883437, 10.05065744493165999680554829956, 11.03904453931258304080450091226