L(s) = 1 | + (1.86 + 1.35i)2-s + (−0.146 − 0.451i)3-s + (1.02 + 3.16i)4-s + (0.338 − 1.04i)6-s − 3.03·7-s + (−0.951 + 2.92i)8-s + (2.24 − 1.63i)9-s + (−1.61 − 1.17i)11-s + (1.28 − 0.930i)12-s + (−1.15 + 0.838i)13-s + (−5.67 − 4.12i)14-s + (−0.357 + 0.259i)16-s + (−0.574 + 1.76i)17-s + 6.40·18-s + (0.279 − 0.859i)19-s + ⋯ |
L(s) = 1 | + (1.32 + 0.959i)2-s + (−0.0847 − 0.260i)3-s + (0.514 + 1.58i)4-s + (0.138 − 0.425i)6-s − 1.14·7-s + (−0.336 + 1.03i)8-s + (0.748 − 0.543i)9-s + (−0.487 − 0.354i)11-s + (0.369 − 0.268i)12-s + (−0.320 + 0.232i)13-s + (−1.51 − 1.10i)14-s + (−0.0893 + 0.0649i)16-s + (−0.139 + 0.429i)17-s + 1.51·18-s + (0.0640 − 0.197i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58293 + 0.882022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58293 + 0.882022i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-1.86 - 1.35i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.146 + 0.451i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.15 - 0.838i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.574 - 1.76i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.279 + 0.859i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.69 + 1.95i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 3.76i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.99 - 6.12i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.09 + 2.24i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.48 + 1.07i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.59T + 43T^{2} \) |
| 47 | \( 1 + (-1.48 - 4.56i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.93 - 9.03i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (8.61 - 6.25i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (11.5 + 8.39i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.30 + 10.1i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.85 - 11.8i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.216 - 0.157i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.64 + 8.15i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.89 + 12.0i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-3.85 - 2.80i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.07 - 9.47i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.56249938877005315111782882775, −12.66914139146565446379561337632, −12.28509488134141291784130608872, −10.52590501377522926186587045972, −9.240409521309884996960382367225, −7.62140112512924906455517114496, −6.68085975050181996319782043361, −5.91426700662497836953546673554, −4.45012197463429773012824138212, −3.20362452341818143024499620600,
2.40386567053213333965417650351, 3.78618392323871594925482783495, 4.91303661089280062182410966095, 6.11547494431231781371395254198, 7.61130565525451402883261522629, 9.677314518738723250195404443638, 10.24731731000774315635563086033, 11.37721254750256987995062106886, 12.45050532133126147799590362306, 13.11929915097082875510031993134