| L(s) = 1 | + 2-s − 1.60·3-s + 4-s − 1.76·5-s − 1.60·6-s − 0.621·7-s + 8-s − 0.436·9-s − 1.76·10-s + 5.06·11-s − 1.60·12-s + 6.63·13-s − 0.621·14-s + 2.82·15-s + 16-s + 6.40·17-s − 0.436·18-s − 6.97·19-s − 1.76·20-s + 0.995·21-s + 5.06·22-s + 1.76·23-s − 1.60·24-s − 1.88·25-s + 6.63·26-s + 5.50·27-s − 0.621·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.924·3-s + 0.5·4-s − 0.789·5-s − 0.653·6-s − 0.234·7-s + 0.353·8-s − 0.145·9-s − 0.558·10-s + 1.52·11-s − 0.462·12-s + 1.84·13-s − 0.166·14-s + 0.729·15-s + 0.250·16-s + 1.55·17-s − 0.102·18-s − 1.59·19-s − 0.394·20-s + 0.217·21-s + 1.07·22-s + 0.367·23-s − 0.326·24-s − 0.377·25-s + 1.30·26-s + 1.05·27-s − 0.117·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.142504974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142504974\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3019 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.60T + 3T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 + 0.621T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 - 0.133T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 2.39T + 41T^{2} \) |
| 43 | \( 1 + 3.33T + 43T^{2} \) |
| 47 | \( 1 - 2.14T + 47T^{2} \) |
| 53 | \( 1 - 0.508T + 53T^{2} \) |
| 59 | \( 1 + 7.92T + 59T^{2} \) |
| 61 | \( 1 - 2.52T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 2.75T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 9.15T + 89T^{2} \) |
| 97 | \( 1 + 1.95T + 97T^{2} \) |
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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
−8.193025477150381392143436387177, −7.00441972093044344459042221716, −6.41294357503161228448076326221, −6.07480062533918057602438622185, −5.26669977875535444297212036725, −4.36148580466009699270368587775, −3.68280153979380526619726240102, −3.26356943780599244882306969066, −1.68000995179195345767407390035, −0.76433272538389610018470799856,
0.76433272538389610018470799856, 1.68000995179195345767407390035, 3.26356943780599244882306969066, 3.68280153979380526619726240102, 4.36148580466009699270368587775, 5.26669977875535444297212036725, 6.07480062533918057602438622185, 6.41294357503161228448076326221, 7.00441972093044344459042221716, 8.193025477150381392143436387177
