| L(s) = 1 | + 3-s − 3·7-s + 9-s − 3·11-s + 13-s + 3·17-s − 3·21-s − 3·23-s + 27-s + 8·29-s − 4·31-s − 3·33-s + 37-s + 39-s − 3·41-s + 4·43-s + 10·47-s + 2·49-s + 3·51-s − 9·53-s − 4·59-s + 9·61-s − 3·63-s − 4·67-s − 3·69-s − 7·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.727·17-s − 0.654·21-s − 0.625·23-s + 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.522·33-s + 0.164·37-s + 0.160·39-s − 0.468·41-s + 0.609·43-s + 1.45·47-s + 2/7·49-s + 0.420·51-s − 1.23·53-s − 0.520·59-s + 1.15·61-s − 0.377·63-s − 0.488·67-s − 0.361·69-s − 0.830·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−16.10591036071937, −15.81248106354314, −15.30621542470686, −14.48987367985963, −14.09840727785148, −13.42403050270740, −13.05044977335105, −12.39051022287205, −12.06175662012692, −11.10849257872200, −10.42210838372031, −10.07191466268516, −9.489241689755455, −8.850065587523136, −8.249032518955277, −7.629243166718028, −7.090518365392122, −6.281357944632358, −5.818760638433299, −4.996959574541512, −4.199854708957315, −3.439020855314875, −2.923178447925555, −2.239289723760594, −1.121785889743790, 0,
1.121785889743790, 2.239289723760594, 2.923178447925555, 3.439020855314875, 4.199854708957315, 4.996959574541512, 5.818760638433299, 6.281357944632358, 7.090518365392122, 7.629243166718028, 8.249032518955277, 8.850065587523136, 9.489241689755455, 10.07191466268516, 10.42210838372031, 11.10849257872200, 12.06175662012692, 12.39051022287205, 13.05044977335105, 13.42403050270740, 14.09840727785148, 14.48987367985963, 15.30621542470686, 15.81248106354314, 16.10591036071937
