L(s) = 1 | + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)9-s + (1.76 − 0.896i)13-s + (−0.278 − 1.76i)17-s + (−0.309 − 0.951i)25-s + (−0.5 + 0.363i)29-s + (0.809 + 1.58i)37-s + (0.363 − 1.11i)41-s + (0.809 − 0.587i)45-s − i·49-s + (0.309 + 0.0489i)53-s + (1.53 − 0.5i)61-s + (−0.309 + 1.95i)65-s + (1.76 + 0.896i)73-s + (0.809 + 0.587i)81-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)5-s + (−0.951 − 0.309i)9-s + (1.76 − 0.896i)13-s + (−0.278 − 1.76i)17-s + (−0.309 − 0.951i)25-s + (−0.5 + 0.363i)29-s + (0.809 + 1.58i)37-s + (0.363 − 1.11i)41-s + (0.809 − 0.587i)45-s − i·49-s + (0.309 + 0.0489i)53-s + (1.53 − 0.5i)61-s + (−0.309 + 1.95i)65-s + (1.76 + 0.896i)73-s + (0.809 + 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.031809867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.031809867\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
good | 3 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.278 + 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \) |
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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
−8.586172837859195374764935973421, −8.167100797930879268692217980926, −7.23982243917906902942949120683, −6.56443041623214789759677076574, −5.82484557196755966205407976371, −5.04381964245982842898416246984, −3.81246579954198126158675274481, −3.24754200186197610132636050923, −2.47664765871470159988706684440, −0.72785290252308125464931330632,
1.19697567233760887196401608385, 2.26630454388658692102564065705, 3.77967290164660695972365977510, 3.96127964812492233976519614987, 5.10878437146811207599040779400, 6.00551719610959777270606485849, 6.40549396296497669456561903543, 7.76575590326420346921401202641, 8.167494162809066777289958127026, 8.914206415050773209144503505020