L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.222 − 0.385i)3-s + (−0.499 − 0.866i)4-s + (0.222 + 0.385i)6-s + 0.999·8-s + (0.400 + 0.694i)9-s + (−0.623 + 1.07i)11-s − 0.445·12-s + (−0.5 + 0.866i)16-s + (−0.623 − 1.07i)17-s − 0.801·18-s + (0.900 + 1.56i)19-s + (−0.623 − 1.07i)22-s + (0.222 − 0.385i)24-s + 25-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.222 − 0.385i)3-s + (−0.499 − 0.866i)4-s + (0.222 + 0.385i)6-s + 0.999·8-s + (0.400 + 0.694i)9-s + (−0.623 + 1.07i)11-s − 0.445·12-s + (−0.5 + 0.866i)16-s + (−0.623 − 1.07i)17-s − 0.801·18-s + (0.900 + 1.56i)19-s + (−0.623 − 1.07i)22-s + (0.222 − 0.385i)24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8622980754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8622980754\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 89 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.834977756181250338127690651114, −9.093105647682295263568142798235, −8.116679680877874895759408149414, −7.44209345525168841785914643570, −7.09179651743395116653107996380, −5.92008424662041937196990938214, −5.03017690675187409264107863164, −4.33524516052654345595041026373, −2.61646363051807880001575732542, −1.45920643869141690856520438350,
0.956653224392413600006933524198, 2.57277238392896110891284874963, 3.35061092365312034518835297000, 4.25804046699410206982416124806, 5.20562048363748483562654979979, 6.48544032319112908366029012587, 7.38082964008895199993455267010, 8.436000473376787286909301001334, 8.899375355198828034118291298548, 9.630487298225092251300628574792