L(s) = 1 | + (0.809 − 0.587i)5-s + (0.309 + 0.951i)9-s + (−0.5 − 1.53i)13-s + (−1.11 − 1.53i)17-s + (0.309 − 0.951i)25-s + (1.11 − 1.53i)29-s + (−0.190 − 0.587i)37-s + (0.5 + 1.53i)41-s + (0.809 + 0.587i)45-s − 49-s + (1.30 + 0.951i)53-s + (1.11 + 0.363i)61-s + (−1.30 − 0.951i)65-s + (1.11 + 0.363i)73-s + (−0.809 + 0.587i)81-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)5-s + (0.309 + 0.951i)9-s + (−0.5 − 1.53i)13-s + (−1.11 − 1.53i)17-s + (0.309 − 0.951i)25-s + (1.11 − 1.53i)29-s + (−0.190 − 0.587i)37-s + (0.5 + 1.53i)41-s + (0.809 + 0.587i)45-s − 49-s + (1.30 + 0.951i)53-s + (1.11 + 0.363i)61-s + (−1.30 − 0.951i)65-s + (1.11 + 0.363i)73-s + (−0.809 + 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.378033230\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378033230\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
good | 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−8.650728082530714320097532382401, −8.039466362402280562274495082458, −7.32186567973350642971300539965, −6.42431328856794322968208566587, −5.53586724180165001563099815103, −4.94692469092921468492526216909, −4.33501812463327090338104655637, −2.74967141423410022935365826485, −2.32541086982479454478523979869, −0.855779514923479521528128194326,
1.55793908799703878535171724715, 2.30000879745002115212267968323, 3.48899782055941207676087143385, 4.22575677171895238233258594378, 5.16886497011220838195987559863, 6.21874137446378448677823162967, 6.72013944302832404364797835358, 7.09032652804719987969349769062, 8.486541551166683915719739743608, 8.967292774498838364294699795550