L(s) = 1 | − 0.858·2-s + (1.60 + 0.639i)3-s − 1.26·4-s + 2.44i·5-s + (−1.38 − 0.549i)6-s − i·7-s + 2.80·8-s + (2.18 + 2.05i)9-s − 2.10i·10-s + (−3.19 + 0.883i)11-s + (−2.03 − 0.807i)12-s + 5.20i·13-s + 0.858i·14-s + (−1.56 + 3.93i)15-s + 0.121·16-s + 0.151·17-s + ⋯ |
L(s) = 1 | − 0.607·2-s + (0.929 + 0.369i)3-s − 0.631·4-s + 1.09i·5-s + (−0.564 − 0.224i)6-s − 0.377i·7-s + 0.990·8-s + (0.727 + 0.686i)9-s − 0.664i·10-s + (−0.963 + 0.266i)11-s + (−0.586 − 0.233i)12-s + 1.44i·13-s + 0.229i·14-s + (−0.404 + 1.01i)15-s + 0.0303·16-s + 0.0367·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.736853 + 0.660921i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.736853 + 0.660921i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.60 - 0.639i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (3.19 - 0.883i)T \) |
good | 2 | \( 1 + 0.858T + 2T^{2} \) |
| 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 13 | \( 1 - 5.20iT - 13T^{2} \) |
| 17 | \( 1 - 0.151T + 17T^{2} \) |
| 19 | \( 1 - 3.38iT - 19T^{2} \) |
| 23 | \( 1 + 5.75iT - 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 5.46T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 2.83T + 41T^{2} \) |
| 43 | \( 1 + 9.44iT - 43T^{2} \) |
| 47 | \( 1 + 0.242iT - 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 2.22iT - 59T^{2} \) |
| 61 | \( 1 + 11.1iT - 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 + 7.57iT - 71T^{2} \) |
| 73 | \( 1 - 3.49iT - 73T^{2} \) |
| 79 | \( 1 - 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 2.58iT - 89T^{2} \) |
| 97 | \( 1 + 5.06T + 97T^{2} \) |
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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
−12.58991684756089806186728981411, −10.93910707726691248115024974918, −10.30988786512695525528325498297, −9.576194907943516264388337124386, −8.542171034157081905609610912949, −7.67676896657413871564758246018, −6.74814900469536910950841254611, −4.78871401190560469938064964657, −3.75323016016836443040769332717, −2.24697676654091503666568136951,
0.990742756082754741049501171350, 2.92963405560250436927423347351, 4.56606102278166294002226250580, 5.61461253124997088012701433807, 7.60301186429814031204619254318, 8.136039981610187423227662260558, 8.964449551334843711663883112764, 9.644480968680631584892640622944, 10.72812959335873851620954100882, 12.33258002311079853618175414378