| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s − 2·14-s − 16-s − 2·17-s − 2·19-s + 8·23-s − 2·28-s − 2·29-s + 2·31-s − 5·32-s + 2·34-s + 8·37-s + 2·38-s − 2·41-s − 4·43-s − 8·46-s − 4·47-s − 3·49-s − 6·53-s + 6·56-s + 2·58-s + 12·59-s + 10·61-s − 2·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s − 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.458·19-s + 1.66·23-s − 0.377·28-s − 0.371·29-s + 0.359·31-s − 0.883·32-s + 0.342·34-s + 1.31·37-s + 0.324·38-s − 0.312·41-s − 0.609·43-s − 1.17·46-s − 0.583·47-s − 3/7·49-s − 0.824·53-s + 0.801·56-s + 0.262·58-s + 1.56·59-s + 1.28·61-s − 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.91944725702772, −14.64155826059151, −14.21767418232188, −13.34110824633352, −13.09453838401422, −12.74252416784656, −11.68853774718781, −11.34423544917458, −10.92765537088966, −10.21424334876263, −9.833038839311203, −9.146836810797947, −8.697892141048046, −8.279488743891003, −7.726268713657654, −7.103089095341203, −6.595969349884863, −5.724354093164964, −5.052468694915222, −4.628724353146555, −4.079600053775651, −3.227995905712113, −2.407346320075187, −1.580105106598561, −0.9738840865051881, 0,
0.9738840865051881, 1.580105106598561, 2.407346320075187, 3.227995905712113, 4.079600053775651, 4.628724353146555, 5.052468694915222, 5.724354093164964, 6.595969349884863, 7.103089095341203, 7.726268713657654, 8.279488743891003, 8.697892141048046, 9.146836810797947, 9.833038839311203, 10.21424334876263, 10.92765537088966, 11.34423544917458, 11.68853774718781, 12.74252416784656, 13.09453838401422, 13.34110824633352, 14.21767418232188, 14.64155826059151, 14.91944725702772
