L(s) = 1 | + (−0.654 + 0.755i)3-s + (−0.959 + 0.281i)4-s + (0.273 + 1.89i)7-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)12-s + (−0.118 + 0.258i)13-s + (0.841 − 0.540i)16-s + (0.186 − 1.29i)19-s + (−1.61 − 1.03i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−0.797 − 1.74i)28-s + (0.698 + 1.53i)31-s + (0.415 + 0.909i)36-s + 0.830·37-s + ⋯ |
L(s) = 1 | + (−0.654 + 0.755i)3-s + (−0.959 + 0.281i)4-s + (0.273 + 1.89i)7-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)12-s + (−0.118 + 0.258i)13-s + (0.841 − 0.540i)16-s + (0.186 − 1.29i)19-s + (−1.61 − 1.03i)21-s + (0.415 − 0.909i)25-s + (0.841 + 0.540i)27-s + (−0.797 − 1.74i)28-s + (0.698 + 1.53i)31-s + (0.415 + 0.909i)36-s + 0.830·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0231 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4854696331\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4854696331\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (-0.415 + 0.909i)T \) |
good | 2 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 5 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (-0.273 - 1.89i)T + (-0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 17 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
| 23 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 37 | \( 1 - 0.830T + T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 61 | \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 71 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \) |
| 83 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 89 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 97 | \( 1 + 1.30T + T^{2} \) |
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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
−12.56454392178166689227560628310, −12.07019487061140072682560788389, −11.09821257820483302099802542808, −9.772470639335758558595474273014, −9.030112322810735742308218180929, −8.366334632644317498082118206967, −6.46649899546939721231958118624, −5.26967697284962259688452034107, −4.65588446674150082276570924847, −2.96749018349298029463062836871,
1.14623430820223783389319663343, 3.89623354500837733503281158679, 4.96842016623834166123314345611, 6.20322432546940861793431699772, 7.48134284303205171297777550253, 8.103434766751510728251708567167, 9.773069753235315718726476912626, 10.49383983029239418321288854902, 11.40336053937425555435787078790, 12.69117910975437728205348444729