L(s) = 1 | − 2-s + 4-s + (−0.707 + 0.707i)7-s − 8-s + (−0.707 − 0.292i)11-s + (0.707 − 0.707i)14-s + 16-s + (0.707 + 0.292i)22-s + (0.707 + 0.707i)25-s + (−0.707 + 0.707i)28-s + (0.707 + 1.70i)29-s − 32-s + (−0.292 + 0.707i)37-s + (−0.707 + 1.70i)43-s + (−0.707 − 0.292i)44-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + (−0.707 + 0.707i)7-s − 8-s + (−0.707 − 0.292i)11-s + (0.707 − 0.707i)14-s + 16-s + (0.707 + 0.292i)22-s + (0.707 + 0.707i)25-s + (−0.707 + 0.707i)28-s + (0.707 + 1.70i)29-s − 32-s + (−0.292 + 0.707i)37-s + (−0.707 + 1.70i)43-s + (−0.707 − 0.292i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0264 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0264 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5093990801\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5093990801\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (-1 - i)T + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 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
−9.461658481328248174090984676554, −8.766339738068245598098670543149, −8.199164758542848781621179467126, −7.24247715040740272877557093936, −6.55159505199474326552679683291, −5.76829299546126172531517036531, −4.89943604328929353183270299990, −3.23249212489918434301172589328, −2.78735109666548780952418628151, −1.40997760053730715258220385359,
0.51000993293398624026820823225, 2.08544524374370391571382113657, 3.02960876366254441446318137034, 4.06922833846949171414156592637, 5.27161458509778567789529720085, 6.31337914067575404348146795501, 6.89141021847263827147911690026, 7.68299463919269769924648854440, 8.342588076033095564907000907914, 9.180108367044822599313656518818