L(s) = 1 | + (−0.245 − 0.969i)2-s − 0.649i·3-s + (−0.879 + 0.475i)4-s − 1.83i·5-s + (−0.629 + 0.159i)6-s + (0.677 + 0.735i)8-s + 0.578·9-s + (−1.77 + 0.449i)10-s + 0.329i·11-s + (0.309 + 0.571i)12-s − 1.18·15-s + (0.546 − 0.837i)16-s − 1.09·17-s + (−0.141 − 0.560i)18-s + (0.871 + 1.61i)20-s + ⋯ |
L(s) = 1 | + (−0.245 − 0.969i)2-s − 0.649i·3-s + (−0.879 + 0.475i)4-s − 1.83i·5-s + (−0.629 + 0.159i)6-s + (0.677 + 0.735i)8-s + 0.578·9-s + (−1.77 + 0.449i)10-s + 0.329i·11-s + (0.309 + 0.571i)12-s − 1.18·15-s + (0.546 − 0.837i)16-s − 1.09·17-s + (−0.141 − 0.560i)18-s + (0.871 + 1.61i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7436974711\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7436974711\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.245 + 0.969i)T \) |
| 359 | \( 1 + T \) |
good | 3 | \( 1 + 0.649iT - T^{2} \) |
| 5 | \( 1 + 1.83iT - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.329iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + 1.09T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + 1.75T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.99iT - T^{2} \) |
| 41 | \( 1 + 0.165T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 1.57T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.490T + T^{2} \) |
| 79 | \( 1 - 0.803T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.526521275164567666928599919449, −8.016275861452499639440590404645, −7.26455048582189088212933768925, −6.06698640452842648696789121047, −5.13651897007615494511032846619, −4.35670539275448965592921699857, −3.90083997712648203798869044577, −2.15774378866898994505230756276, −1.69991618003679776808011601461, −0.50347692938117134674454567475,
1.96474740131150746514095683532, 3.25240561477521632175488386114, 3.99688305080165402270848422573, 4.77674262280104237722342551284, 5.91576055397878032130723478571, 6.52395229924627897022917614714, 7.01478633050042079900255379973, 7.84354726343949569457381649944, 8.520620175268250198750487388966, 9.657830559567978981768640663759