| L(s) = 1 | − 3-s − 3·5-s + 7-s − 2·9-s + 3·11-s − 13-s + 3·15-s + 19-s − 21-s + 23-s + 4·25-s + 5·27-s + 6·29-s − 4·31-s − 3·33-s − 3·35-s + 10·37-s + 39-s + 10·43-s + 6·45-s − 6·47-s + 49-s + 12·53-s − 9·55-s − 57-s + 6·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 0.774·15-s + 0.229·19-s − 0.218·21-s + 0.208·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s − 0.522·33-s − 0.507·35-s + 1.64·37-s + 0.160·39-s + 1.52·43-s + 0.894·45-s − 0.875·47-s + 1/7·49-s + 1.64·53-s − 1.21·55-s − 0.132·57-s + 0.781·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.80348282892404, −13.08961889150820, −12.50357592052838, −12.17509507302401, −11.62052471567800, −11.43787799671444, −11.05891031602770, −10.46008642347393, −9.897524129311831, −9.175491661977913, −8.837168594830818, −8.251307246034050, −7.805266063400679, −7.390141825243026, −6.642526773915683, −6.449884003903822, −5.486244031743130, −5.347748447178071, −4.397511949169021, −4.203732297306912, −3.613514007713401, −2.841162988057831, −2.404617570110937, −1.254740321839445, −0.7904946285811711, 0,
0.7904946285811711, 1.254740321839445, 2.404617570110937, 2.841162988057831, 3.613514007713401, 4.203732297306912, 4.397511949169021, 5.347748447178071, 5.486244031743130, 6.449884003903822, 6.642526773915683, 7.390141825243026, 7.805266063400679, 8.251307246034050, 8.837168594830818, 9.175491661977913, 9.897524129311831, 10.46008642347393, 11.05891031602770, 11.43787799671444, 11.62052471567800, 12.17509507302401, 12.50357592052838, 13.08961889150820, 13.80348282892404
