L(s) = 1 | + (−0.587 + 0.809i)9-s + (−0.142 − 0.896i)13-s + (0.278 + 0.142i)17-s + (1.11 + 0.363i)29-s + (1.95 − 0.309i)37-s + (1.53 + 1.11i)41-s + i·49-s + (1.58 − 0.809i)53-s + (−1.53 + 1.11i)61-s + (−1.76 − 0.278i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.896 + 1.76i)97-s + 1.61·101-s + (0.363 − 0.5i)109-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)9-s + (−0.142 − 0.896i)13-s + (0.278 + 0.142i)17-s + (1.11 + 0.363i)29-s + (1.95 − 0.309i)37-s + (1.53 + 1.11i)41-s + i·49-s + (1.58 − 0.809i)53-s + (−1.53 + 1.11i)61-s + (−1.76 − 0.278i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.896 + 1.76i)97-s + 1.61·101-s + (0.363 − 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.243683497\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.243683497\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.278 - 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.95 + 0.309i)T + (0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-1.58 + 0.809i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
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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
−8.600612902783382737166569245059, −7.82050157150394267143334467161, −7.50408943150717824527051177474, −6.27392303197630440052101944047, −5.77435490952757076617706843744, −4.92717321185023965981026197802, −4.21005358235834226903620952262, −2.99578827622824067597478935302, −2.48985672056177143594054329864, −1.06279979876293503187293731204,
0.876628085897858628262699573574, 2.25216586337053787055338854721, 3.08297345838531656918322175391, 4.05922371059531065298397089313, 4.70889562720184078455647837515, 5.84094598483132373083210550006, 6.25159723909029149179616191432, 7.14166683074017238291324011103, 7.81290366635244655285951402117, 8.768936956863952426856705783828