| L(s) = 1 | − 2·5-s − 2·7-s − 3·9-s + 2·11-s + 13-s + 6·17-s + 6·19-s + 8·23-s − 25-s − 2·29-s + 10·31-s + 4·35-s + 6·37-s − 6·41-s − 4·43-s + 6·45-s − 2·47-s − 3·49-s − 6·53-s − 4·55-s + 10·59-s + 2·61-s + 6·63-s − 2·65-s − 10·67-s + 10·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 9-s + 0.603·11-s + 0.277·13-s + 1.45·17-s + 1.37·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.894·45-s − 0.291·47-s − 3/7·49-s − 0.824·53-s − 0.539·55-s + 1.30·59-s + 0.256·61-s + 0.755·63-s − 0.248·65-s − 1.22·67-s + 1.18·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.195693820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195693820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.06947661821394918418145315819, −9.420900365687786965896429939417, −8.448574941224861101937906300558, −7.73101944712663475124809392314, −6.78863087763573008439633106061, −5.85741842949549920807230123970, −4.88282077350781318055546355500, −3.47757941400193154474288876226, −3.07083233977996346947903399516, −0.906662202456000794708098813478,
0.906662202456000794708098813478, 3.07083233977996346947903399516, 3.47757941400193154474288876226, 4.88282077350781318055546355500, 5.85741842949549920807230123970, 6.78863087763573008439633106061, 7.73101944712663475124809392314, 8.448574941224861101937906300558, 9.420900365687786965896429939417, 10.06947661821394918418145315819
