L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (1.52 − 1.63i)5-s − 0.999·6-s + (−0.381 − 0.220i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.507 − 2.17i)10-s − 4.82·11-s + (−0.866 + 0.499i)12-s + (−5.17 − 2.98i)13-s − 0.440·14-s + (−2.13 + 0.649i)15-s + (−0.5 − 0.866i)16-s + (−4.52 + 2.61i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.683 − 0.730i)5-s − 0.408·6-s + (−0.144 − 0.0832i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.160 − 0.688i)10-s − 1.45·11-s + (−0.249 + 0.144i)12-s + (−1.43 − 0.829i)13-s − 0.117·14-s + (−0.552 + 0.167i)15-s + (−0.125 − 0.216i)16-s + (−1.09 + 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9891093310\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9891093310\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-1.52 + 1.63i)T \) |
| 37 | \( 1 + (4.28 + 4.31i)T \) |
good | 7 | \( 1 + (0.381 + 0.220i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + (5.17 + 2.98i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (4.52 - 2.61i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.49 - 4.31i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.03iT - 23T^{2} \) |
| 29 | \( 1 - 6.01T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 41 | \( 1 + (-2.74 + 4.75i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 3.34iT - 43T^{2} \) |
| 47 | \( 1 - 4.75iT - 47T^{2} \) |
| 53 | \( 1 + (9.44 - 5.45i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.43 + 7.69i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.63 + 8.03i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.414 - 0.239i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.95 - 5.11i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 9.65iT - 73T^{2} \) |
| 79 | \( 1 + (0.379 - 0.657i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.36 - 1.36i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.0225 - 0.0390i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.810587055587552990913141716590, −8.468801303427127567287395137777, −7.82083806546290071732512650762, −6.57368094317068566728543753578, −5.90674932851852883342057641200, −4.95810444332050375374363353475, −4.53928993702244578440985467231, −2.79753681408832133350222959797, −2.01105967481734331644561536086, −0.33599262391336300725299400607,
2.35690362992655231988794747469, 2.92135203521438821025196307906, 4.68002724308637702634407212495, 4.91345595916994457292908416161, 6.09644398935074477736036244043, 6.80709073421775762122617813669, 7.40341143848131774148594115076, 8.626537451462114439996007483409, 9.697696869311804971911657822094, 10.21845659046396770537199451552