L(s) = 1 | + (0.341 + 1.74i)2-s + (−2.00 + 0.815i)4-s + (−0.461 − 0.887i)7-s + (−1.13 − 1.73i)8-s + (−0.739 + 0.673i)9-s + (−0.000667 − 0.121i)11-s + (1.39 − 1.10i)14-s + (1.09 − 1.06i)16-s + (−1.42 − 1.06i)18-s + (0.211 − 0.0425i)22-s + (−1.65 − 0.448i)23-s + (−0.795 + 0.605i)25-s + (1.64 + 1.40i)28-s + (−0.0415 − 1.50i)29-s + (0.527 + 0.366i)32-s + ⋯ |
L(s) = 1 | + (0.341 + 1.74i)2-s + (−2.00 + 0.815i)4-s + (−0.461 − 0.887i)7-s + (−1.13 − 1.73i)8-s + (−0.739 + 0.673i)9-s + (−0.000667 − 0.121i)11-s + (1.39 − 1.10i)14-s + (1.09 − 1.06i)16-s + (−1.42 − 1.06i)18-s + (0.211 − 0.0425i)22-s + (−1.65 − 0.448i)23-s + (−0.795 + 0.605i)25-s + (1.64 + 1.40i)28-s + (−0.0415 − 1.50i)29-s + (0.527 + 0.366i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2626400568\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2626400568\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.461 + 0.887i)T \) |
| 571 | \( 1 + (-0.716 + 0.697i)T \) |
good | 2 | \( 1 + (-0.341 - 1.74i)T + (-0.926 + 0.376i)T^{2} \) |
| 3 | \( 1 + (0.739 - 0.673i)T^{2} \) |
| 5 | \( 1 + (0.795 - 0.605i)T^{2} \) |
| 11 | \( 1 + (0.000667 + 0.121i)T + (-0.999 + 0.0110i)T^{2} \) |
| 13 | \( 1 + (0.709 + 0.705i)T^{2} \) |
| 17 | \( 1 + (-0.959 + 0.282i)T^{2} \) |
| 19 | \( 1 + (-0.202 + 0.979i)T^{2} \) |
| 23 | \( 1 + (1.65 + 0.448i)T + (0.863 + 0.504i)T^{2} \) |
| 29 | \( 1 + (0.0415 + 1.50i)T + (-0.998 + 0.0550i)T^{2} \) |
| 31 | \( 1 + (0.879 - 0.475i)T^{2} \) |
| 37 | \( 1 + (0.977 - 0.0756i)T + (0.988 - 0.153i)T^{2} \) |
| 41 | \( 1 + (0.821 + 0.569i)T^{2} \) |
| 43 | \( 1 + (-0.719 + 1.59i)T + (-0.660 - 0.750i)T^{2} \) |
| 47 | \( 1 + (-0.851 - 0.523i)T^{2} \) |
| 53 | \( 1 + (-1.39 + 1.29i)T + (0.0715 - 0.997i)T^{2} \) |
| 59 | \( 1 + (-0.945 + 0.324i)T^{2} \) |
| 61 | \( 1 + (-0.471 - 0.882i)T^{2} \) |
| 67 | \( 1 + (0.0859 + 0.373i)T + (-0.899 + 0.436i)T^{2} \) |
| 71 | \( 1 + (1.46 + 0.310i)T + (0.913 + 0.406i)T^{2} \) |
| 73 | \( 1 + (0.360 - 0.932i)T^{2} \) |
| 79 | \( 1 + (1.77 - 0.863i)T + (0.618 - 0.785i)T^{2} \) |
| 83 | \( 1 + (0.997 - 0.0770i)T^{2} \) |
| 89 | \( 1 + (-0.565 - 0.824i)T^{2} \) |
| 97 | \( 1 + (0.441 + 0.897i)T^{2} \) |
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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
−8.339441920676303994064748963875, −7.61404676851766301537321551860, −7.22179305422697188066141957974, −6.24167491747301118222323118584, −5.83845767935967839352254644828, −5.06773837747527141984417234991, −4.12772503492095030858790009492, −3.65142727525881319762397150474, −2.23200034575736678024829347672, −0.12638775975786622149061136615,
1.48306963084392906427555337055, 2.43251570347684529757939005045, 3.10380144748638864769154506783, 3.81195467122257448804390647889, 4.63292588103729675061794707504, 5.72797423871067861271862956364, 5.97828561494506181110701855794, 7.24472292611746704847163741306, 8.467554610396870925708266298247, 8.861971370786442764482451388291