| L(s) = 1 | − 3·2-s + 4-s + 9·5-s + 2·7-s + 21·8-s − 27·10-s − 30·11-s − 6·14-s − 71·16-s + 111·17-s − 46·19-s + 9·20-s + 90·22-s + 6·23-s − 44·25-s + 2·28-s + 105·29-s − 100·31-s + 45·32-s − 333·34-s + 18·35-s + 17·37-s + 138·38-s + 189·40-s + 231·41-s − 514·43-s − 30·44-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s + 0.804·5-s + 0.107·7-s + 0.928·8-s − 0.853·10-s − 0.822·11-s − 0.114·14-s − 1.10·16-s + 1.58·17-s − 0.555·19-s + 0.100·20-s + 0.872·22-s + 0.0543·23-s − 0.351·25-s + 0.0134·28-s + 0.672·29-s − 0.579·31-s + 0.248·32-s − 1.67·34-s + 0.0869·35-s + 0.0755·37-s + 0.589·38-s + 0.747·40-s + 0.879·41-s − 1.82·43-s − 0.102·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 17 | \( 1 - 111 T + p^{3} T^{2} \) |
| 19 | \( 1 + 46 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 T + p^{3} T^{2} \) |
| 29 | \( 1 - 105 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 17 T + p^{3} T^{2} \) |
| 41 | \( 1 - 231 T + p^{3} T^{2} \) |
| 43 | \( 1 + 514 T + p^{3} T^{2} \) |
| 47 | \( 1 - 162 T + p^{3} T^{2} \) |
| 53 | \( 1 + 639 T + p^{3} T^{2} \) |
| 59 | \( 1 + 600 T + p^{3} T^{2} \) |
| 61 | \( 1 - 233 T + p^{3} T^{2} \) |
| 67 | \( 1 - 926 T + p^{3} T^{2} \) |
| 71 | \( 1 - 930 T + p^{3} T^{2} \) |
| 73 | \( 1 + 253 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1324 T + p^{3} T^{2} \) |
| 83 | \( 1 + 810 T + p^{3} T^{2} \) |
| 89 | \( 1 + 498 T + p^{3} T^{2} \) |
| 97 | \( 1 - 14 p T + p^{3} T^{2} \) |
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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
−8.682972386261179174557038859205, −8.038263293687297801116545838310, −7.41643940349391247560926189412, −6.32589935209881622990059876288, −5.42873358453323535741815580340, −4.66475584683334183300667883871, −3.33846548891653449816670222871, −2.12091531081045564354298857925, −1.22140671015271970789027629264, 0,
1.22140671015271970789027629264, 2.12091531081045564354298857925, 3.33846548891653449816670222871, 4.66475584683334183300667883871, 5.42873358453323535741815580340, 6.32589935209881622990059876288, 7.41643940349391247560926189412, 8.038263293687297801116545838310, 8.682972386261179174557038859205
