L(s) = 1 | + (−0.285 + 0.495i)2-s + (0.285 − 1.70i)3-s + (0.836 + 1.44i)4-s + (−0.5 − 0.866i)5-s + (0.764 + 0.630i)6-s + (−0.714 + 1.23i)7-s − 2.10·8-s + (−2.83 − 0.977i)9-s + 0.571·10-s + (−1.33 + 2.31i)11-s + (2.71 − 1.01i)12-s + (−2.33 − 4.04i)13-s + (−0.408 − 0.707i)14-s + (−1.62 + 0.606i)15-s + (−1.07 + 1.85i)16-s + 2.67·17-s + ⋯ |
L(s) = 1 | + (−0.202 + 0.350i)2-s + (0.165 − 0.986i)3-s + (0.418 + 0.724i)4-s + (−0.223 − 0.387i)5-s + (0.312 + 0.257i)6-s + (−0.269 + 0.467i)7-s − 0.742·8-s + (−0.945 − 0.325i)9-s + 0.180·10-s + (−0.402 + 0.697i)11-s + (0.783 − 0.292i)12-s + (−0.648 − 1.12i)13-s + (−0.109 − 0.189i)14-s + (−0.418 + 0.156i)15-s + (−0.267 + 0.464i)16-s + 0.648·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.759796 + 0.00657468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.759796 + 0.00657468i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.285 + 1.70i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.285 - 0.495i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (0.714 - 1.23i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.33 - 2.31i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.33 + 4.04i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 + (-2.95 - 5.12i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.74 + 8.21i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.48 + 6.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + (-0.735 - 1.27i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.235 - 0.408i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.47 - 6.02i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + (-0.571 - 0.990i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.26 + 2.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.29 - 5.70i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.71T + 73T^{2} \) |
| 79 | \( 1 + (-0.143 + 0.249i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.14 + 3.71i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + (3.91 - 6.78i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−15.86743016413490047852388663725, −14.97130758299893278305717536386, −13.26539609170015479572141596592, −12.43425909057846734327392994612, −11.64981260226710242967313892980, −9.517499571512560640742638575240, −7.997343931752726173805196531392, −7.38995516787053244561912536982, −5.70657316268924195173900432208, −2.88630941288728393362215202062,
3.14319303985762875120891141004, 5.16955999748238025204935450543, 6.90406840563948206761509656245, 8.893951282230359066206896472447, 10.10105042524175183232538873076, 10.83542569523987005041963807753, 11.95441576373610528950077135057, 14.05913989684097082650737061262, 14.65782820857765487869413824386, 15.99891136939233648747677686354