L(s) = 1 | + 3-s + 5-s + 7-s − 2·9-s + 4·11-s − 5·13-s + 15-s + 4·17-s − 19-s + 21-s − 23-s − 4·25-s − 5·27-s − 4·31-s + 4·33-s + 35-s − 10·37-s − 5·39-s − 6·41-s + 8·43-s − 2·45-s + 4·47-s + 49-s + 4·51-s + 8·53-s + 4·55-s − 57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 1.20·11-s − 1.38·13-s + 0.258·15-s + 0.970·17-s − 0.229·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s − 0.962·27-s − 0.718·31-s + 0.696·33-s + 0.169·35-s − 1.64·37-s − 0.800·39-s − 0.937·41-s + 1.21·43-s − 0.298·45-s + 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.09·53-s + 0.539·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.71592779914715, −14.31772872355629, −13.96915957539377, −13.56743283025635, −12.70586047231423, −12.21745809420828, −11.82712381573981, −11.38519870087240, −10.58933496753595, −10.06195351402962, −9.578507375623528, −9.105563351611046, −8.581182872424439, −8.084134056122656, −7.304883410550549, −7.094316954364407, −6.202915493378768, −5.579672743720493, −5.258632392932450, −4.390885949961945, −3.723989802486130, −3.258643310857457, −2.270882161424684, −2.036974821137330, −1.086643819934247, 0,
1.086643819934247, 2.036974821137330, 2.270882161424684, 3.258643310857457, 3.723989802486130, 4.390885949961945, 5.258632392932450, 5.579672743720493, 6.202915493378768, 7.094316954364407, 7.304883410550549, 8.084134056122656, 8.581182872424439, 9.105563351611046, 9.578507375623528, 10.06195351402962, 10.58933496753595, 11.38519870087240, 11.82712381573981, 12.21745809420828, 12.70586047231423, 13.56743283025635, 13.96915957539377, 14.31772872355629, 14.71592779914715