| L(s) = 1 | + 0.311·2-s − 2.52·3-s − 1.90·4-s − 5-s − 0.785·6-s − 7-s − 1.21·8-s + 3.37·9-s − 0.311·10-s − 3.52·11-s + 4.80·12-s − 5.49·13-s − 0.311·14-s + 2.52·15-s + 3.42·16-s − 1.37·17-s + 1.05·18-s + 1.90·20-s + 2.52·21-s − 1.09·22-s − 7.95·23-s + 3.06·24-s + 25-s − 1.70·26-s − 0.954·27-s + 1.90·28-s + 5.52·29-s + ⋯ |
| L(s) = 1 | + 0.219·2-s − 1.45·3-s − 0.951·4-s − 0.447·5-s − 0.320·6-s − 0.377·7-s − 0.429·8-s + 1.12·9-s − 0.0983·10-s − 1.06·11-s + 1.38·12-s − 1.52·13-s − 0.0831·14-s + 0.652·15-s + 0.857·16-s − 0.334·17-s + 0.247·18-s + 0.425·20-s + 0.551·21-s − 0.233·22-s − 1.65·23-s + 0.625·24-s + 0.200·25-s − 0.335·26-s − 0.183·27-s + 0.359·28-s + 1.02·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 0.311T + 2T^{2} \) |
| 3 | \( 1 + 2.52T + 3T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 - 5.52T + 29T^{2} \) |
| 31 | \( 1 - 8.57T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 - 9.80T + 43T^{2} \) |
| 47 | \( 1 + 4.90T + 47T^{2} \) |
| 53 | \( 1 - 2.90T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 - 3.92T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 0.0967T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 0.857T + 97T^{2} \) |
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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
−7.46217108735824276034466639242, −6.59254758951534054162174782555, −6.01523619910755824254598573838, −5.19836502226413686711330885015, −4.83965149078881245261450835155, −4.26126207750268709073780212731, −3.24285142235842102399217411629, −2.28098958063844800708929632714, −0.65523664896309391042862243809, 0,
0.65523664896309391042862243809, 2.28098958063844800708929632714, 3.24285142235842102399217411629, 4.26126207750268709073780212731, 4.83965149078881245261450835155, 5.19836502226413686711330885015, 6.01523619910755824254598573838, 6.59254758951534054162174782555, 7.46217108735824276034466639242
