L(s) = 1 | − 2·5-s + 11-s + 2·13-s + 2·17-s − 4·19-s + 6·23-s − 25-s + 6·29-s − 10·31-s + 2·37-s − 2·41-s + 10·43-s + 6·53-s − 2·55-s − 12·59-s − 2·61-s − 4·65-s − 12·67-s − 2·71-s − 2·73-s − 14·83-s − 4·85-s + 10·89-s + 8·95-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.301·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s + 1.11·29-s − 1.79·31-s + 0.328·37-s − 0.312·41-s + 1.52·43-s + 0.824·53-s − 0.269·55-s − 1.56·59-s − 0.256·61-s − 0.496·65-s − 1.46·67-s − 0.237·71-s − 0.234·73-s − 1.53·83-s − 0.433·85-s + 1.05·89-s + 0.820·95-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.669397893\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.669397893\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.86401908362311, −14.37664005299347, −13.84991119538777, −13.13965894861698, −12.72985139004330, −12.20426213831176, −11.70353332145116, −11.07398144330789, −10.78527112727169, −10.17692858265780, −9.416987808471111, −8.729747109380197, −8.656016774828360, −7.607547030412235, −7.482789953397855, −6.746526593031282, −6.035450690865006, −5.614464619163172, −4.601268156102320, −4.357384654892732, −3.507685168977923, −3.132324281166917, −2.175349944275337, −1.328185863889487, −0.5049791445860975,
0.5049791445860975, 1.328185863889487, 2.175349944275337, 3.132324281166917, 3.507685168977923, 4.357384654892732, 4.601268156102320, 5.614464619163172, 6.035450690865006, 6.746526593031282, 7.482789953397855, 7.607547030412235, 8.656016774828360, 8.729747109380197, 9.416987808471111, 10.17692858265780, 10.78527112727169, 11.07398144330789, 11.70353332145116, 12.20426213831176, 12.72985139004330, 13.13965894861698, 13.84991119538777, 14.37664005299347, 14.86401908362311