| L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s − 4·13-s − 3·15-s + 7·17-s − 6·19-s − 21-s + 4·23-s + 4·25-s + 27-s + 2·29-s + 8·31-s + 3·35-s + 6·37-s − 4·39-s − 6·41-s − 9·43-s − 3·45-s − 3·47-s + 49-s + 7·51-s + 6·53-s − 6·57-s + 3·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.774·15-s + 1.69·17-s − 1.37·19-s − 0.218·21-s + 0.834·23-s + 4/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.507·35-s + 0.986·37-s − 0.640·39-s − 0.937·41-s − 1.37·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s + 0.980·51-s + 0.824·53-s − 0.794·57-s + 0.390·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.31704521040547, −13.05507307727580, −12.49996428608119, −12.07089746889039, −11.66819184406189, −11.36442707716960, −10.40406600591639, −10.10908968376546, −9.907709764477782, −9.040127014449940, −8.571709295818806, −8.211492474500365, −7.728027931422590, −7.263141183963036, −6.875170509422681, −6.246669982704268, −5.606511554618212, −4.777918058342232, −4.579914398714198, −3.926320629244283, −3.310749071533678, −2.964269685363817, −2.388489059639316, −1.487705501448943, −0.7359844806849728, 0,
0.7359844806849728, 1.487705501448943, 2.388489059639316, 2.964269685363817, 3.310749071533678, 3.926320629244283, 4.579914398714198, 4.777918058342232, 5.606511554618212, 6.246669982704268, 6.875170509422681, 7.263141183963036, 7.728027931422590, 8.211492474500365, 8.571709295818806, 9.040127014449940, 9.907709764477782, 10.10908968376546, 10.40406600591639, 11.36442707716960, 11.66819184406189, 12.07089746889039, 12.49996428608119, 13.05507307727580, 13.31704521040547
