L(s) = 1 | + (−0.891 + 0.453i)3-s + i·4-s + (1.10 + 0.678i)7-s + (0.587 − 0.809i)9-s + (−0.453 − 0.891i)12-s + (0.119 + 0.101i)13-s − 16-s + (−1.40 + 0.581i)19-s + (−1.29 − 0.101i)21-s + (0.587 + 0.809i)25-s + (−0.156 + 0.987i)27-s + (−0.678 + 1.10i)28-s + (0.178 + 0.744i)31-s + (0.809 + 0.587i)36-s + (0.794 − 0.678i)37-s + ⋯ |
L(s) = 1 | + (−0.891 + 0.453i)3-s + i·4-s + (1.10 + 0.678i)7-s + (0.587 − 0.809i)9-s + (−0.453 − 0.891i)12-s + (0.119 + 0.101i)13-s − 16-s + (−1.40 + 0.581i)19-s + (−1.29 − 0.101i)21-s + (0.587 + 0.809i)25-s + (−0.156 + 0.987i)27-s + (−0.678 + 1.10i)28-s + (0.178 + 0.744i)31-s + (0.809 + 0.587i)36-s + (0.794 − 0.678i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0817 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0817 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7716929238\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7716929238\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.891 - 0.453i)T \) |
| 241 | \( 1 + (0.891 - 0.453i)T \) |
good | 2 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (-1.10 - 0.678i)T + (0.453 + 0.891i)T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.119 - 0.101i)T + (0.156 + 0.987i)T^{2} \) |
| 17 | \( 1 + (0.891 - 0.453i)T^{2} \) |
| 19 | \( 1 + (1.40 - 0.581i)T + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.987 + 0.156i)T^{2} \) |
| 29 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 31 | \( 1 + (-0.178 - 0.744i)T + (-0.891 + 0.453i)T^{2} \) |
| 37 | \( 1 + (-0.794 + 0.678i)T + (0.156 - 0.987i)T^{2} \) |
| 41 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 43 | \( 1 + (1.01 + 1.65i)T + (-0.453 + 0.891i)T^{2} \) |
| 47 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 61 | \( 1 + (0.533 + 1.04i)T + (-0.587 + 0.809i)T^{2} \) |
| 67 | \( 1 + (-1.87 - 0.297i)T + (0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (-0.453 + 0.891i)T^{2} \) |
| 73 | \( 1 + (0.355 - 0.303i)T + (0.156 - 0.987i)T^{2} \) |
| 79 | \( 1 + (0.280 - 0.550i)T + (-0.587 - 0.809i)T^{2} \) |
| 83 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 97 | \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−11.11951647867829258631039189951, −10.13522008364931495249269625787, −8.909760272490771673139785701845, −8.427918146669724283954347342337, −7.34840319677435545766726690842, −6.40666428271411134011395790422, −5.31935475000001675500800220688, −4.52408111059706711549852476155, −3.55419314435999164149829661703, −1.99465812269000676796825668164,
1.01271942470610658630927416191, 2.18529203568086938100292936089, 4.47276930001084073944882899195, 4.83672008349479713335453484206, 6.06464926050435510047230666982, 6.64278376436962419704114948983, 7.69703502059571623871248520025, 8.556009321180918711157321412218, 9.856178569229682505693897391662, 10.59236604923278222734838679230