L(s) = 1 | + 2-s − 2.85·3-s + 4-s + 5-s − 2.85·6-s − 7-s + 8-s + 5.17·9-s + 10-s − 2.85·12-s − 2.17·13-s − 14-s − 2.85·15-s + 16-s − 3.85·17-s + 5.17·18-s − 1.68·19-s + 20-s + 2.85·21-s + 6.85·23-s − 2.85·24-s + 25-s − 2.17·26-s − 6.22·27-s − 28-s + 2·29-s − 2.85·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.65·3-s + 0.5·4-s + 0.447·5-s − 1.16·6-s − 0.377·7-s + 0.353·8-s + 1.72·9-s + 0.316·10-s − 0.825·12-s − 0.603·13-s − 0.267·14-s − 0.738·15-s + 0.250·16-s − 0.936·17-s + 1.22·18-s − 0.386·19-s + 0.223·20-s + 0.623·21-s + 1.43·23-s − 0.583·24-s + 0.200·25-s − 0.426·26-s − 1.19·27-s − 0.188·28-s + 0.371·29-s − 0.522·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.472139201\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.472139201\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.85T + 3T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 0.682T + 31T^{2} \) |
| 37 | \( 1 + 3.71T + 37T^{2} \) |
| 41 | \( 1 + 9.03T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 7.75T + 59T^{2} \) |
| 61 | \( 1 + 8.89T + 61T^{2} \) |
| 67 | \( 1 - 9.85T + 67T^{2} \) |
| 71 | \( 1 + 0.542T + 71T^{2} \) |
| 73 | \( 1 - 5.85T + 73T^{2} \) |
| 79 | \( 1 + 0.317T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 + 0.682T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−7.25738439198942551912173886094, −6.81898185348300023123208352977, −6.37333130429951713446547988452, −5.65508667957368090107541290007, −4.98167215629574662728184175044, −4.67725694195606627204744319315, −3.66993567305326112179020337661, −2.67066530771975070837000507294, −1.71558816597654519640782976052, −0.57705017119737752831639128591,
0.57705017119737752831639128591, 1.71558816597654519640782976052, 2.67066530771975070837000507294, 3.66993567305326112179020337661, 4.67725694195606627204744319315, 4.98167215629574662728184175044, 5.65508667957368090107541290007, 6.37333130429951713446547988452, 6.81898185348300023123208352977, 7.25738439198942551912173886094