| L(s) = 1 | + 1.69·5-s − 7-s − 2.47·11-s + 0.777·13-s − 2.81·17-s − 4.98·19-s − 0.712·23-s − 2.11·25-s − 4.51·29-s + 5.09·31-s − 1.69·35-s − 6.87·37-s + 5.87·41-s − 4.65·43-s − 12.9·47-s + 49-s − 1.88·53-s − 4.21·55-s − 14.2·59-s + 14.3·61-s + 1.32·65-s + 7.98·67-s + 10.2·71-s + 4.98·73-s + 2.47·77-s − 9.21·79-s − 8.81·83-s + ⋯ |
| L(s) = 1 | + 0.760·5-s − 0.377·7-s − 0.746·11-s + 0.215·13-s − 0.681·17-s − 1.14·19-s − 0.148·23-s − 0.422·25-s − 0.837·29-s + 0.915·31-s − 0.287·35-s − 1.13·37-s + 0.917·41-s − 0.709·43-s − 1.89·47-s + 0.142·49-s − 0.259·53-s − 0.567·55-s − 1.86·59-s + 1.83·61-s + 0.163·65-s + 0.975·67-s + 1.21·71-s + 0.583·73-s + 0.282·77-s − 1.03·79-s − 0.967·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 - 0.777T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 + 0.712T + 23T^{2} \) |
| 29 | \( 1 + 4.51T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 + 4.65T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 1.88T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 7.98T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 4.98T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 - 8.64T + 97T^{2} \) |
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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
−8.587126446621447996136384147269, −8.024938131524604605797188585875, −6.90606317598642338959280255309, −6.31461797549043900918047929759, −5.54461936363834807667104636575, −4.68581710803585717343852613335, −3.67947540686004779507345532516, −2.57770546725762967941170523160, −1.76303607186285460965851604880, 0,
1.76303607186285460965851604880, 2.57770546725762967941170523160, 3.67947540686004779507345532516, 4.68581710803585717343852613335, 5.54461936363834807667104636575, 6.31461797549043900918047929759, 6.90606317598642338959280255309, 8.024938131524604605797188585875, 8.587126446621447996136384147269
