| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s − 2·13-s + 14-s − 16-s + 8·19-s − 5·25-s + 2·26-s + 28-s − 2·29-s − 5·32-s − 8·38-s + 6·41-s − 8·43-s − 8·47-s + 49-s + 5·50-s + 2·52-s + 8·53-s − 3·56-s + 2·58-s − 4·59-s − 8·61-s + 7·64-s + 8·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s − 0.554·13-s + 0.267·14-s − 1/4·16-s + 1.83·19-s − 25-s + 0.392·26-s + 0.188·28-s − 0.371·29-s − 0.883·32-s − 1.29·38-s + 0.937·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.707·50-s + 0.277·52-s + 1.09·53-s − 0.400·56-s + 0.262·58-s − 0.520·59-s − 1.02·61-s + 7/8·64-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.31387599187931, −14.73975690169566, −14.13690451345335, −13.64472751276290, −13.29691941353108, −12.68701600359394, −11.94976863019940, −11.65670602377609, −10.88439284502938, −10.28065654881997, −9.792518845626236, −9.393543740557994, −9.023261784910897, −8.172405897713971, −7.694120379317037, −7.367538656547364, −6.575819988445182, −5.872506803622422, −5.115360267071796, −4.829965062024739, −3.816329117980165, −3.438665657369759, −2.498056519330628, −1.655959513222900, −0.8562900200678915, 0,
0.8562900200678915, 1.655959513222900, 2.498056519330628, 3.438665657369759, 3.816329117980165, 4.829965062024739, 5.115360267071796, 5.872506803622422, 6.575819988445182, 7.367538656547364, 7.694120379317037, 8.172405897713971, 9.023261784910897, 9.393543740557994, 9.792518845626236, 10.28065654881997, 10.88439284502938, 11.65670602377609, 11.94976863019940, 12.68701600359394, 13.29691941353108, 13.64472751276290, 14.13690451345335, 14.73975690169566, 15.31387599187931
