L(s) = 1 | + 3·5-s + 5·7-s + 11-s + 2·13-s + 17-s + 19-s − 4·23-s + 4·25-s + 2·29-s + 6·31-s + 15·35-s + 43-s − 9·47-s + 18·49-s − 10·53-s + 3·55-s − 8·59-s − 61-s + 6·65-s − 8·67-s − 12·71-s − 11·73-s + 5·77-s − 16·79-s + 12·83-s + 3·85-s + 6·89-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 1.88·7-s + 0.301·11-s + 0.554·13-s + 0.242·17-s + 0.229·19-s − 0.834·23-s + 4/5·25-s + 0.371·29-s + 1.07·31-s + 2.53·35-s + 0.152·43-s − 1.31·47-s + 18/7·49-s − 1.37·53-s + 0.404·55-s − 1.04·59-s − 0.128·61-s + 0.744·65-s − 0.977·67-s − 1.42·71-s − 1.28·73-s + 0.569·77-s − 1.80·79-s + 1.31·83-s + 0.325·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.207472030\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.207472030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.742321760001961651389979974377, −8.160007849857414045140648086964, −7.43632371515983923998772054443, −6.29636417830616150950587212845, −5.80065514036351948381475791738, −4.91760568249510581017872290537, −4.32680757840779686690441622638, −2.94683125587980903660881898164, −1.81200784323266148009001357293, −1.34947676285250801422046885118,
1.34947676285250801422046885118, 1.81200784323266148009001357293, 2.94683125587980903660881898164, 4.32680757840779686690441622638, 4.91760568249510581017872290537, 5.80065514036351948381475791738, 6.29636417830616150950587212845, 7.43632371515983923998772054443, 8.160007849857414045140648086964, 8.742321760001961651389979974377