L(s) = 1 | − 2·5-s + 7-s + 2·11-s + 2·13-s − 6·17-s + 4·19-s + 6·23-s − 25-s + 4·31-s − 2·35-s + 10·37-s − 2·41-s + 4·43-s + 4·47-s + 49-s + 12·53-s − 4·55-s + 12·59-s + 6·61-s − 4·65-s + 4·67-s − 14·71-s − 2·73-s + 2·77-s + 8·79-s − 16·83-s + 12·85-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.603·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.25·23-s − 1/5·25-s + 0.718·31-s − 0.338·35-s + 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 1.64·53-s − 0.539·55-s + 1.56·59-s + 0.768·61-s − 0.496·65-s + 0.488·67-s − 1.66·71-s − 0.234·73-s + 0.227·77-s + 0.900·79-s − 1.75·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.470879677\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.470879677\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.930312173970229319450826669816, −8.962490139668736518272616433529, −8.399008275227548308212304898711, −7.41496908072272826170175484750, −6.74586296479860441225382306846, −5.64648595445333773791578057156, −4.50269149377754204048383750620, −3.85404415863451499904477094777, −2.59482356264800310932574632605, −0.986854704293823535020818033237,
0.986854704293823535020818033237, 2.59482356264800310932574632605, 3.85404415863451499904477094777, 4.50269149377754204048383750620, 5.64648595445333773791578057156, 6.74586296479860441225382306846, 7.41496908072272826170175484750, 8.399008275227548308212304898711, 8.962490139668736518272616433529, 9.930312173970229319450826669816