L(s) = 1 | + (0.707 − 1.58i)3-s − 1.41i·5-s + i·7-s + (−2.00 − 2.23i)9-s − 4.47·11-s − 7.16·13-s + (−2.23 − 1.00i)15-s − 7.30i·17-s + 0.837i·19-s + (1.58 + 0.707i)21-s + 5.65·23-s + 2.99·25-s + (−4.94 + 1.58i)27-s − 1.64i·29-s + 6.32i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.912i)3-s − 0.632i·5-s + 0.377i·7-s + (−0.666 − 0.745i)9-s − 1.34·11-s − 1.98·13-s + (−0.577 − 0.258i)15-s − 1.77i·17-s + 0.192i·19-s + (0.345 + 0.154i)21-s + 1.17·23-s + 0.599·25-s + (−0.952 + 0.304i)27-s − 0.305i·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.178320 - 0.966597i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.178320 - 0.966597i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 1.58i)T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 7.16T + 13T^{2} \) |
| 17 | \( 1 + 7.30iT - 17T^{2} \) |
| 19 | \( 1 - 0.837iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 1.64iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 4.32T + 37T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + 8.32iT - 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 1.18iT - 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 + 1.18iT - 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.897303144573287517927530047090, −9.179305283288325857641616406767, −8.341012768643968675024422656487, −7.39591315237539197024622410086, −6.94599380835986193236998139339, −5.26740138267717700547329139967, −5.00624298894617178978774223207, −2.97557803021906792295090887207, −2.29858261603519240425404158228, −0.45951354822095963822983817039,
2.41562304396615535181400716163, 3.17653944485991332451710318583, 4.52151204689432941265831724335, 5.17575738597635070062296597413, 6.46939106857623855430117744011, 7.64955527945946435504999753575, 8.126245974680700130480785585077, 9.441170933425563387469955594813, 10.05457660438176651028779921522, 10.71830857802738436694821934283