L(s) = 1 | + 2-s + (−0.5 − 1.53i)3-s + 4-s + (−0.5 − 1.53i)6-s + 8-s + (−1.30 + 0.951i)9-s + (0.190 + 0.587i)11-s + (−0.5 − 1.53i)12-s + 16-s + (0.618 − 1.90i)17-s + (−1.30 + 0.951i)18-s + (−1.61 − 1.17i)19-s + (0.190 + 0.587i)22-s + (−0.5 − 1.53i)24-s + 25-s + ⋯ |
L(s) = 1 | + 2-s + (−0.5 − 1.53i)3-s + 4-s + (−0.5 − 1.53i)6-s + 8-s + (−1.30 + 0.951i)9-s + (0.190 + 0.587i)11-s + (−0.5 − 1.53i)12-s + 16-s + (0.618 − 1.90i)17-s + (−1.30 + 0.951i)18-s + (−1.61 − 1.17i)19-s + (0.190 + 0.587i)22-s + (−0.5 − 1.53i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.855694768\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.855694768\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 251 | \( 1 + (0.809 - 0.587i)T \) |
good | 3 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 11 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−8.981893290305750694354372118728, −7.933762312276523046538951497032, −7.20872443976209079613900885115, −6.75735513830182829714032355311, −6.13776501237703708797100137191, −5.08780076158096774088830567224, −4.53255540667671884101917998738, −2.96872576047878490064981224139, −2.28945656651594676027275666223, −1.10018684094431658516196510905,
1.89151238285185649856963161542, 3.43639917452265306101856813983, 3.82629587850914389270770732306, 4.58402567721183794783402779099, 5.55442959367847410153223555095, 5.98320338710540756900179282016, 6.80846697561595552973360330109, 8.208733157342159440285173067057, 8.687446837726954302103030513209, 9.964961833105443904181461007031