L(s) = 1 | + (−2.22 + 3.84i)5-s + (1.45 + 2.51i)7-s + (−1.08 − 1.87i)11-s + (1.96 − 3.40i)13-s − 1.79·17-s − 1.76·19-s + (3.44 − 5.96i)23-s + (−7.36 − 12.7i)25-s + (−2.87 − 4.97i)29-s + (3.27 − 5.67i)31-s − 12.9·35-s + 2.51·37-s + (3.68 − 6.38i)41-s + (−2.53 − 4.39i)43-s + (4.98 + 8.63i)47-s + ⋯ |
L(s) = 1 | + (−0.993 + 1.71i)5-s + (0.549 + 0.952i)7-s + (−0.326 − 0.565i)11-s + (0.544 − 0.943i)13-s − 0.435·17-s − 0.405·19-s + (0.717 − 1.24i)23-s + (−1.47 − 2.54i)25-s + (−0.533 − 0.924i)29-s + (0.588 − 1.01i)31-s − 2.18·35-s + 0.413·37-s + (0.575 − 0.996i)41-s + (−0.386 − 0.669i)43-s + (0.727 + 1.25i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.082740934\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082740934\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.22 - 3.84i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.45 - 2.51i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.08 + 1.87i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.96 + 3.40i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + (-3.44 + 5.96i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.87 + 4.97i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.27 + 5.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 + (-3.68 + 6.38i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.53 + 4.39i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.98 - 8.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 3.30T + 53T^{2} \) |
| 59 | \( 1 + (2.30 - 3.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.87 + 3.24i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.36 - 4.08i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.907T + 71T^{2} \) |
| 73 | \( 1 + 1.87T + 73T^{2} \) |
| 79 | \( 1 + (-1.23 - 2.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.09 + 1.89i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.30T + 89T^{2} \) |
| 97 | \( 1 + (-4.45 - 7.71i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.272321085564563692421215209507, −7.893791673632573376732747047723, −7.09754480540722359140825069150, −6.19641083874466474506243852174, −5.79111615136289096278658043968, −4.53531099948370114557753873123, −3.74178479753379285146775892127, −2.76725058598340132033155320741, −2.39580414457560962054156025723, −0.37896173648944961791902943566,
1.05982438616860946576956151134, 1.68508336391776050089633126595, 3.43959620674220740616339068638, 4.22910928017543884964011533077, 4.70535144987671209337123566914, 5.30371777888396218229837480106, 6.56456329840776549732618654014, 7.47834672684470576948447439823, 7.81624429143256514709850792289, 8.797595933968058132009316959326