L(s) = 1 | + (0.623 + 0.781i)2-s + (1.62 − 0.781i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + (1.62 + 0.781i)6-s + (−0.222 + 0.974i)7-s + (−0.900 + 0.433i)8-s + (1.40 − 1.75i)9-s + (−0.900 − 0.433i)10-s + (0.400 + 1.75i)12-s + (−0.900 + 0.433i)14-s + (−1.12 + 1.40i)15-s + (−0.900 − 0.433i)16-s + 2.24·18-s + (−0.222 − 0.974i)20-s + (0.400 + 1.75i)21-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (1.62 − 0.781i)3-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)5-s + (1.62 + 0.781i)6-s + (−0.222 + 0.974i)7-s + (−0.900 + 0.433i)8-s + (1.40 − 1.75i)9-s + (−0.900 − 0.433i)10-s + (0.400 + 1.75i)12-s + (−0.900 + 0.433i)14-s + (−1.12 + 1.40i)15-s + (−0.900 − 0.433i)16-s + 2.24·18-s + (−0.222 − 0.974i)20-s + (0.400 + 1.75i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.870469151\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.870469151\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.900 - 0.433i)T \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
good | 3 | \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 11 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 13 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 41 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 61 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 89 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.07383128921799904750127815938, −8.879525555719870868731839018042, −8.481183223515175483637476775458, −7.937536906987108764939375691255, −6.82038336457902720161639508463, −6.63307449290438684940680967977, −5.06032587901650125974617238416, −3.86171520246635756029948128972, −3.08714242012451760878389346611, −2.34567439231546085324130294657,
1.61867505132137259139394609582, 3.17211905749499924936273074096, 3.57959460580788791764157760725, 4.39222377080146198078585319136, 5.13427281381952993493234755769, 6.86493640063060644502461403434, 7.81919594904862576422927641068, 8.508105378245217952621060142487, 9.586272044808339363792093875439, 9.799315691051631517175964389303