L(s) = 1 | + 2-s + 2.01·3-s + 4-s − 3.34·5-s + 2.01·6-s − 2.94·7-s + 8-s + 1.04·9-s − 3.34·10-s − 4.10·11-s + 2.01·12-s + 0.449·13-s − 2.94·14-s − 6.73·15-s + 16-s + 1.49·17-s + 1.04·18-s − 7.57·19-s − 3.34·20-s − 5.92·21-s − 4.10·22-s − 23-s + 2.01·24-s + 6.21·25-s + 0.449·26-s − 3.92·27-s − 2.94·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.16·3-s + 0.5·4-s − 1.49·5-s + 0.821·6-s − 1.11·7-s + 0.353·8-s + 0.349·9-s − 1.05·10-s − 1.23·11-s + 0.580·12-s + 0.124·13-s − 0.787·14-s − 1.73·15-s + 0.250·16-s + 0.362·17-s + 0.247·18-s − 1.73·19-s − 0.748·20-s − 1.29·21-s − 0.875·22-s − 0.208·23-s + 0.410·24-s + 1.24·25-s + 0.0881·26-s − 0.755·27-s − 0.556·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 2.01T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 + 2.94T + 7T^{2} \) |
| 11 | \( 1 + 4.10T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 + 7.57T + 19T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 - 5.92T + 41T^{2} \) |
| 43 | \( 1 + 8.61T + 43T^{2} \) |
| 47 | \( 1 + 7.72T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 7.90T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 - 1.93T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 + 4.60T + 73T^{2} \) |
| 79 | \( 1 - 9.25T + 79T^{2} \) |
| 83 | \( 1 - 3.54T + 83T^{2} \) |
| 89 | \( 1 - 4.12T + 89T^{2} \) |
| 97 | \( 1 + 8.83T + 97T^{2} \) |
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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
−9.030541719088440242764768755286, −8.139834006558116048288553715413, −7.81456706483696464087400377248, −6.84864456587286158609841368482, −5.91956188937387166435867816306, −4.62131959410934031803016828689, −3.81078655063144176209246689863, −3.14879594824268961438391376911, −2.38837131003762306788926428518, 0,
2.38837131003762306788926428518, 3.14879594824268961438391376911, 3.81078655063144176209246689863, 4.62131959410934031803016828689, 5.91956188937387166435867816306, 6.84864456587286158609841368482, 7.81456706483696464087400377248, 8.139834006558116048288553715413, 9.030541719088440242764768755286