L(s) = 1 | + 2-s + 4-s + 5-s − 3.46·7-s + 8-s + 10-s + 2.73·11-s − 1.26·13-s − 3.46·14-s + 16-s + 4·17-s + 5.46·19-s + 20-s + 2.73·22-s + 0.535·23-s + 25-s − 1.26·26-s − 3.46·28-s − 0.732·29-s − 31-s + 32-s + 4·34-s − 3.46·35-s − 6.73·37-s + 5.46·38-s + 40-s + 2.53·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.30·7-s + 0.353·8-s + 0.316·10-s + 0.823·11-s − 0.351·13-s − 0.925·14-s + 0.250·16-s + 0.970·17-s + 1.25·19-s + 0.223·20-s + 0.582·22-s + 0.111·23-s + 0.200·25-s − 0.248·26-s − 0.654·28-s − 0.135·29-s − 0.179·31-s + 0.176·32-s + 0.685·34-s − 0.585·35-s − 1.10·37-s + 0.886·38-s + 0.158·40-s + 0.396·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.986190234\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.986190234\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 0.535T + 23T^{2} \) |
| 29 | \( 1 + 0.732T + 29T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 3.26T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−9.055408332272148111790290815368, −7.84285933033472996730914795427, −7.04192158454520368531120960630, −6.48191669578323408484533670509, −5.67636326008322642073216233427, −5.08069528569622461021649513319, −3.79705868053443139565495142759, −3.33209770690154324381907122019, −2.33468802284947989128665926626, −0.998153172715650978517550824087,
0.998153172715650978517550824087, 2.33468802284947989128665926626, 3.33209770690154324381907122019, 3.79705868053443139565495142759, 5.08069528569622461021649513319, 5.67636326008322642073216233427, 6.48191669578323408484533670509, 7.04192158454520368531120960630, 7.84285933033472996730914795427, 9.055408332272148111790290815368