L(s) = 1 | + (−0.929 − 0.368i)3-s + (−0.809 + 0.587i)4-s + (0.939 − 0.516i)7-s + (0.728 + 0.684i)9-s + (0.968 − 0.248i)12-s + (0.781 + 1.23i)13-s + (0.309 − 0.951i)16-s + (1.60 − 1.16i)19-s + (−1.06 + 0.134i)21-s + (0.0627 + 0.998i)25-s + (−0.425 − 0.904i)27-s + (−0.456 + 0.969i)28-s + (−0.328 + 1.72i)31-s + (−0.992 − 0.125i)36-s + (−0.996 − 1.57i)37-s + ⋯ |
L(s) = 1 | + (−0.929 − 0.368i)3-s + (−0.809 + 0.587i)4-s + (0.939 − 0.516i)7-s + (0.728 + 0.684i)9-s + (0.968 − 0.248i)12-s + (0.781 + 1.23i)13-s + (0.309 − 0.951i)16-s + (1.60 − 1.16i)19-s + (−1.06 + 0.134i)21-s + (0.0627 + 0.998i)25-s + (−0.425 − 0.904i)27-s + (−0.456 + 0.969i)28-s + (−0.328 + 1.72i)31-s + (−0.992 − 0.125i)36-s + (−0.996 − 1.57i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6332264299\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6332264299\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.929 + 0.368i)T \) |
| 151 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.0627 - 0.998i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 0.516i)T + (0.535 - 0.844i)T^{2} \) |
| 11 | \( 1 + (-0.728 + 0.684i)T^{2} \) |
| 13 | \( 1 + (-0.781 - 1.23i)T + (-0.425 + 0.904i)T^{2} \) |
| 17 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 19 | \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 31 | \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \) |
| 37 | \( 1 + (0.996 + 1.57i)T + (-0.425 + 0.904i)T^{2} \) |
| 41 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 43 | \( 1 + (1.11 + 0.614i)T + (0.535 + 0.844i)T^{2} \) |
| 47 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 53 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.574 - 0.227i)T + (0.728 - 0.684i)T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.32i)T + (0.0627 - 0.998i)T^{2} \) |
| 71 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 73 | \( 1 + (1.41 - 0.779i)T + (0.535 - 0.844i)T^{2} \) |
| 79 | \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \) |
| 83 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 89 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 97 | \( 1 + (-0.450 + 0.423i)T + (0.0627 - 0.998i)T^{2} \) |
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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
−11.40667379260090968607750221731, −10.69329891000101176933602308391, −9.411947740299008134362443072754, −8.626744189464013836294344200464, −7.44457308409725082661820029357, −6.94000079793106143006242935555, −5.35439589216540441825908491460, −4.74523039838255427636850081868, −3.60489321477366912366469596847, −1.43317126145868440553404310603,
1.28928219755368866819241527817, 3.59971708848509602968842548607, 4.82071449953914401456718932672, 5.50624217297569567162341725693, 6.17440393803913764553048233822, 7.82208097092806958418154681387, 8.579930438547510397292594424635, 9.818013941739121599124305843137, 10.25206996616956489287980847495, 11.29512292030711526430207302305