L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s − 0.999·6-s − 8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (0.5 + 0.866i)11-s − 13-s − 0.999·15-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)18-s + (0.5 − 0.866i)19-s − 0.999·22-s + (−0.500 − 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s − 0.999·6-s − 8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (0.5 + 0.866i)11-s − 13-s − 0.999·15-s + (0.5 − 0.866i)16-s + (−0.499 − 0.866i)18-s + (0.5 − 0.866i)19-s − 0.999·22-s + (−0.500 − 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8060561877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8060561877\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.732401668569254544707327709242, −9.273523274845355394213591742851, −8.429510783235846122888409036458, −7.47483374876324960241435659138, −7.23667933566299335625434474809, −6.28908313434472923164938686831, −5.10978028550747113663078302098, −4.24374285879161238281362749116, −3.16285706795461451049843885874, −2.49709993097971151098225125605,
0.69954799717679210461111291224, 1.66849467725264650250704312031, 2.81402686424918645504267357614, 3.62668311505512108156905735745, 4.92566712134190430805831260023, 6.01461138727796430549632800623, 6.73767342947322257079539929568, 7.932748783107998785849115123798, 8.338490068019160363418706545999, 9.184183711327551646145649060819