L(s) = 1 | − 2-s + 0.302·3-s + 4-s − 2.30·5-s − 0.302·6-s + 1.30·7-s − 8-s − 2.90·9-s + 2.30·10-s + 11-s + 0.302·12-s − 3.30·13-s − 1.30·14-s − 0.697·15-s + 16-s − 4.60·17-s + 2.90·18-s + 19-s − 2.30·20-s + 0.394·21-s − 22-s − 1.39·23-s − 0.302·24-s + 0.302·25-s + 3.30·26-s − 1.78·27-s + 1.30·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.174·3-s + 0.5·4-s − 1.02·5-s − 0.123·6-s + 0.492·7-s − 0.353·8-s − 0.969·9-s + 0.728·10-s + 0.301·11-s + 0.0874·12-s − 0.916·13-s − 0.348·14-s − 0.180·15-s + 0.250·16-s − 1.11·17-s + 0.685·18-s + 0.229·19-s − 0.514·20-s + 0.0860·21-s − 0.213·22-s − 0.290·23-s − 0.0618·24-s + 0.0605·25-s + 0.647·26-s − 0.344·27-s + 0.246·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.302T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 - 5.21T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 9.69T + 71T^{2} \) |
| 73 | \( 1 + 2.60T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 4.60T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−11.06182501279895325052782927882, −9.635273451298823125456534079198, −8.892517283374611848571406957455, −7.951483734971078151639838276401, −7.41936718711275481373901664616, −6.15915467907456406136168820306, −4.82792745568016670431215353549, −3.56971673834977105145940364136, −2.18198330808258992032302602749, 0,
2.18198330808258992032302602749, 3.56971673834977105145940364136, 4.82792745568016670431215353549, 6.15915467907456406136168820306, 7.41936718711275481373901664616, 7.951483734971078151639838276401, 8.892517283374611848571406957455, 9.635273451298823125456534079198, 11.06182501279895325052782927882