L(s) = 1 | + (−0.866 + 0.5i)2-s + (1.70 + 0.311i)3-s + (0.499 − 0.866i)4-s − 5-s + (−1.63 + 0.582i)6-s + (2.53 + 0.744i)7-s + 0.999i·8-s + (2.80 + 1.06i)9-s + (0.866 − 0.5i)10-s − 0.441i·11-s + (1.12 − 1.32i)12-s + (3.17 − 1.83i)13-s + (−2.57 + 0.624i)14-s + (−1.70 − 0.311i)15-s + (−0.5 − 0.866i)16-s + (0.136 + 0.235i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.983 + 0.179i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.665 + 0.237i)6-s + (0.959 + 0.281i)7-s + 0.353i·8-s + (0.935 + 0.353i)9-s + (0.273 − 0.158i)10-s − 0.133i·11-s + (0.323 − 0.381i)12-s + (0.880 − 0.508i)13-s + (−0.687 + 0.166i)14-s + (−0.439 − 0.0803i)15-s + (−0.125 − 0.216i)16-s + (0.0330 + 0.0571i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58440 + 0.510062i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58440 + 0.510062i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.70 - 0.311i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.53 - 0.744i)T \) |
good | 11 | \( 1 + 0.441iT - 11T^{2} \) |
| 13 | \( 1 + (-3.17 + 1.83i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.136 - 0.235i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.25 + 1.87i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 2.59iT - 23T^{2} \) |
| 29 | \( 1 + (2.38 + 1.37i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.57 - 4.37i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0597 + 0.103i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.93 - 6.81i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.849 + 1.47i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.94 + 6.83i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0822 + 0.0474i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.60 + 2.78i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (11.2 - 6.50i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.268 + 0.465i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.75iT - 71T^{2} \) |
| 73 | \( 1 + (-9.64 + 5.56i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.51 + 2.62i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.29 - 7.43i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.35 - 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.9 + 7.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.62614079071457939207794045151, −9.614299381130055509860052234285, −8.638728177194171948857926377984, −8.254551644556189155459903611727, −7.51669004003649085524608617917, −6.40624223458657046831824041542, −5.12416275333240595648390320053, −4.07838692068821322607476234384, −2.79272266734773033647505354908, −1.42460873702357336916485098648,
1.29283470923669305625254494568, 2.43458719457828878268522770159, 3.79581383774593265705115541992, 4.51098914844113087543254668080, 6.31174394308091598564569886090, 7.34575133221101300027028669768, 8.111053677252589737879858582358, 8.612461292531976255000707023178, 9.506482178145850694745075525631, 10.52228569573772046643429103911