L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.415 + 0.909i)3-s + (0.841 + 0.540i)4-s + (−0.142 − 0.989i)6-s − 1.97i·7-s + (−0.654 − 0.755i)8-s + (−0.654 + 0.755i)9-s − 1.68·11-s + (−0.142 + 0.989i)12-s + (−0.557 + 1.89i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)18-s − 1.51i·19-s + (1.80 − 0.822i)21-s + (1.61 + 0.474i)22-s + ⋯ |
L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.415 + 0.909i)3-s + (0.841 + 0.540i)4-s + (−0.142 − 0.989i)6-s − 1.97i·7-s + (−0.654 − 0.755i)8-s + (−0.654 + 0.755i)9-s − 1.68·11-s + (−0.142 + 0.989i)12-s + (−0.557 + 1.89i)14-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)18-s − 1.51i·19-s + (1.80 − 0.822i)21-s + (1.61 + 0.474i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5148868228\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5148868228\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 1.97iT - T^{2} \) |
| 11 | \( 1 + 1.68T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.51iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.97iT - T^{2} \) |
| 31 | \( 1 - 0.563iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.30T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 0.830T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.81iT - T^{2} \) |
| 97 | \( 1 - 1.68T + T^{2} \) |
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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
−9.302611424241345770355970086252, −8.348953769338626287544343296954, −7.62909206074225577765801013553, −7.31936691943083719936813175683, −6.07488912716716976780414689653, −4.81576449070102669769484784272, −4.08439928970688080009990319060, −3.14691847581597375063419529901, −2.25496321622871098715489189879, −0.44030265948757455079775548392,
1.74375930733563195692064760898, 2.42271863402911549332038774359, 3.19567922589099793609408758770, 5.36277631470945834837586314826, 5.67629876629311926280097224196, 6.45834256057662031163942774033, 7.60562711708745556745353334204, 7.989912278624246693271542861769, 8.688740748717579136383507981997, 9.243370031524866655609397340408