L(s) = 1 | + (0.959 + 0.281i)2-s + (−0.415 + 0.909i)3-s + (0.841 + 0.540i)4-s + (−0.654 + 0.755i)6-s − 1.97i·7-s + (0.654 + 0.755i)8-s + (−0.654 − 0.755i)9-s + 1.68·11-s + (−0.841 + 0.540i)12-s + (0.557 − 1.89i)14-s + (0.415 + 0.909i)16-s + (−0.415 − 0.909i)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 + (−0.654 + 0.755i)6-s − 1.97i·7-s + (0.654 + 0.755i)8-s + (−0.654 − 0.755i)9-s + 1.68·11-s + (−0.841 + 0.540i)12-s + (0.557 − 1.89i)14-s + (0.415 + 0.909i)16-s + (−0.415 − 0.909i)18-s − 1.51i·19-s + (1.80 + 0.822i)21-s + (1.61 + 0.474i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.958872793\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.958872793\) |
\(L(1)\) |
|
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
−9.507287474850514134527800219861, −8.697428825866822979680180883362, −7.51480056463922992045784626040, −6.76053856793963057213086661636, −6.42557549557149964248863702682, −5.09917469767158025614440286511, −4.50726554546678086068595094581, −3.79694616109210008668861654301, −3.26372463061737767402539760377, −1.32144203248496527897549639167,
1.64767060537152511809024877960, 2.20854121060678313978743047521, 3.36202066681548387104077494987, 4.43503058983717600085115372192, 5.55838882674613714818403783293, 6.12647581893979830433123538325, 6.35423453209796289484342548478, 7.63509848211746274761914893506, 8.357156266401357099493317326072, 9.350447577896645197487830110967