L(s) = 1 | + 1.19·3-s + 1.19·7-s − 1.56·9-s + 5.86·11-s + 0.364·13-s − 1.19·17-s + 19-s + 1.43·21-s − 8.23·23-s − 5.46·27-s − 7.86·29-s − 7.30·31-s + 7.03·33-s − 7.13·37-s + 0.436·39-s + 2.43·41-s − 7.39·43-s − 13.7·47-s − 5.56·49-s − 1.43·51-s + 7.39·53-s + 1.19·57-s − 12.8·59-s − 1.30·61-s − 1.87·63-s + 11.9·67-s − 9.86·69-s + ⋯ |
L(s) = 1 | + 0.692·3-s + 0.453·7-s − 0.521·9-s + 1.76·11-s + 0.101·13-s − 0.290·17-s + 0.229·19-s + 0.313·21-s − 1.71·23-s − 1.05·27-s − 1.46·29-s − 1.31·31-s + 1.22·33-s − 1.17·37-s + 0.0699·39-s + 0.380·41-s − 1.12·43-s − 1.99·47-s − 0.794·49-s − 0.201·51-s + 1.01·53-s + 0.158·57-s − 1.67·59-s − 0.166·61-s − 0.236·63-s + 1.45·67-s − 1.18·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 13 | \( 1 - 0.364T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 + 7.13T + 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 + 13.7T + 47T^{2} \) |
| 53 | \( 1 - 7.39T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 + 2.50T + 73T^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 + 3.02T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 + 1.09T + 97T^{2} \) |
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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
−7.64534540236382847063530936782, −6.87645300920565498585604899299, −6.15903779026175051130915760813, −5.51085922211621842519468572854, −4.56479500316353670185119705896, −3.65550151633669366342639790633, −3.42663000976611058933106669918, −1.98582757715842949354529472289, −1.66005377537821495875329695912, 0,
1.66005377537821495875329695912, 1.98582757715842949354529472289, 3.42663000976611058933106669918, 3.65550151633669366342639790633, 4.56479500316353670185119705896, 5.51085922211621842519468572854, 6.15903779026175051130915760813, 6.87645300920565498585604899299, 7.64534540236382847063530936782