| L(s) = 1 | + 2-s − 2.22·3-s + 4-s − 5-s − 2.22·6-s + 7-s + 8-s + 1.93·9-s − 10-s − 4.95·11-s − 2.22·12-s + 1.45·13-s + 14-s + 2.22·15-s + 16-s + 6.89·17-s + 1.93·18-s − 5.33·19-s − 20-s − 2.22·21-s − 4.95·22-s + 3.91·23-s − 2.22·24-s + 25-s + 1.45·26-s + 2.36·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.28·3-s + 0.5·4-s − 0.447·5-s − 0.906·6-s + 0.377·7-s + 0.353·8-s + 0.644·9-s − 0.316·10-s − 1.49·11-s − 0.641·12-s + 0.403·13-s + 0.267·14-s + 0.573·15-s + 0.250·16-s + 1.67·17-s + 0.455·18-s − 1.22·19-s − 0.223·20-s − 0.484·21-s − 1.05·22-s + 0.816·23-s − 0.453·24-s + 0.200·25-s + 0.285·26-s + 0.456·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.394827502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.394827502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| good | 3 | \( 1 + 2.22T + 3T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 + 4.17T + 29T^{2} \) |
| 31 | \( 1 + 6.25T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 - 0.822T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 + 7.06T + 47T^{2} \) |
| 53 | \( 1 - 0.682T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 5.42T + 61T^{2} \) |
| 67 | \( 1 + 3.44T + 67T^{2} \) |
| 73 | \( 1 - 6.66T + 73T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 - 1.97T + 83T^{2} \) |
| 89 | \( 1 - 8.58T + 89T^{2} \) |
| 97 | \( 1 - 6.42T + 97T^{2} \) |
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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
−7.976600903215513562375471705190, −7.46075455942191349424648013722, −6.62291751293746698182824519902, −5.84755949476535182176168587620, −5.24655524058992136779171806228, −4.91021695459759856442798710480, −3.84391818783495138763314203197, −3.06464312882950161106359024483, −1.89803963647944896842413697214, −0.60965342783009242752059979020,
0.60965342783009242752059979020, 1.89803963647944896842413697214, 3.06464312882950161106359024483, 3.84391818783495138763314203197, 4.91021695459759856442798710480, 5.24655524058992136779171806228, 5.84755949476535182176168587620, 6.62291751293746698182824519902, 7.46075455942191349424648013722, 7.976600903215513562375471705190
