| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 3·13-s − 14-s + 16-s + 8·17-s − 5·19-s + 8·23-s + 3·26-s − 28-s − 3·31-s + 32-s + 8·34-s + 37-s − 5·38-s − 6·41-s + 43-s + 8·46-s + 4·47-s + 49-s + 3·52-s + 4·53-s − 56-s − 12·59-s + 5·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s − 1.14·19-s + 1.66·23-s + 0.588·26-s − 0.188·28-s − 0.538·31-s + 0.176·32-s + 1.37·34-s + 0.164·37-s − 0.811·38-s − 0.937·41-s + 0.152·43-s + 1.17·46-s + 0.583·47-s + 1/7·49-s + 0.416·52-s + 0.549·53-s − 0.133·56-s − 1.56·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.003301256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.003301256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.40983251668295, −12.26438701916426, −11.60034002820837, −11.15556030154669, −10.69987790611404, −10.37033020411626, −9.845164921016752, −9.340915431800123, −8.665451812226388, −8.553698104438440, −7.734336918972175, −7.371688067996704, −6.954463988066891, −6.257994789586845, −6.041916527504370, −5.476459597924198, −5.013899341191568, −4.511471720012498, −3.814859254757734, −3.494192752681723, −3.003423589099422, −2.506555564885061, −1.655394939959095, −1.228058818815961, −0.5214012805972619,
0.5214012805972619, 1.228058818815961, 1.655394939959095, 2.506555564885061, 3.003423589099422, 3.494192752681723, 3.814859254757734, 4.511471720012498, 5.013899341191568, 5.476459597924198, 6.041916527504370, 6.257994789586845, 6.954463988066891, 7.371688067996704, 7.734336918972175, 8.553698104438440, 8.665451812226388, 9.340915431800123, 9.845164921016752, 10.37033020411626, 10.69987790611404, 11.15556030154669, 11.60034002820837, 12.26438701916426, 12.40983251668295
