| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 6·13-s + 16-s + 4·17-s + 3·18-s − 4·19-s + 4·23-s − 6·26-s + 2·31-s − 32-s − 4·34-s − 3·36-s + 2·37-s + 4·38-s + 10·41-s + 8·43-s − 4·46-s + 6·52-s + 2·53-s + 14·61-s − 2·62-s + 64-s + 2·67-s + 4·68-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.66·13-s + 1/4·16-s + 0.970·17-s + 0.707·18-s − 0.917·19-s + 0.834·23-s − 1.17·26-s + 0.359·31-s − 0.176·32-s − 0.685·34-s − 1/2·36-s + 0.328·37-s + 0.648·38-s + 1.56·41-s + 1.21·43-s − 0.589·46-s + 0.832·52-s + 0.274·53-s + 1.79·61-s − 0.254·62-s + 1/8·64-s + 0.244·67-s + 0.485·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.959307354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.959307354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.60409458651965, −12.24738981874334, −11.56823814392173, −11.14261751379814, −10.99142410596407, −10.50006855366910, −9.879193599305896, −9.443701554096463, −8.922810276935021, −8.439730472682855, −8.313718329196621, −7.734470733431767, −7.050192193568195, −6.715769100997930, −6.015836948249001, −5.658026960136628, −5.478534937596162, −4.309689948779033, −4.131084536243706, −3.343546999982704, −2.824265553964430, −2.466279065201593, −1.536351243214012, −1.068835572586788, −0.4773067589068978,
0.4773067589068978, 1.068835572586788, 1.536351243214012, 2.466279065201593, 2.824265553964430, 3.343546999982704, 4.131084536243706, 4.309689948779033, 5.478534937596162, 5.658026960136628, 6.015836948249001, 6.715769100997930, 7.050192193568195, 7.734470733431767, 8.313718329196621, 8.439730472682855, 8.922810276935021, 9.443701554096463, 9.879193599305896, 10.50006855366910, 10.99142410596407, 11.14261751379814, 11.56823814392173, 12.24738981874334, 12.60409458651965
