L(s) = 1 | + 2.21·3-s + 5-s − 7-s + 1.90·9-s − 11-s − 2.21·13-s + 2.21·15-s + 4.21·17-s − 7.80·19-s − 2.21·21-s − 2.90·23-s + 25-s − 2.42·27-s + 0.755·29-s − 7.11·31-s − 2.21·33-s − 35-s − 6.28·37-s − 4.90·39-s − 5.54·41-s + 4.14·43-s + 1.90·45-s − 1.03·47-s + 49-s + 9.33·51-s − 6.57·53-s − 55-s + ⋯ |
L(s) = 1 | + 1.27·3-s + 0.447·5-s − 0.377·7-s + 0.634·9-s − 0.301·11-s − 0.614·13-s + 0.571·15-s + 1.02·17-s − 1.79·19-s − 0.483·21-s − 0.605·23-s + 0.200·25-s − 0.467·27-s + 0.140·29-s − 1.27·31-s − 0.385·33-s − 0.169·35-s − 1.03·37-s − 0.785·39-s − 0.866·41-s + 0.632·43-s + 0.283·45-s − 0.150·47-s + 0.142·49-s + 1.30·51-s − 0.903·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 + 7.80T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 + 7.11T + 31T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 + 5.54T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 1.03T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 + 2.68T + 59T^{2} \) |
| 61 | \( 1 - 8.79T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 5.28T + 97T^{2} \) |
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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
−7.74561054309613471995023118514, −7.21306664862750169862317260031, −6.30406766751679723499919846732, −5.62467019608805364693917132863, −4.72430459287232144085620175508, −3.79551181239557047228663755708, −3.17101845672700640940523148081, −2.32350345438549865482109570501, −1.73461352028919521734192270529, 0,
1.73461352028919521734192270529, 2.32350345438549865482109570501, 3.17101845672700640940523148081, 3.79551181239557047228663755708, 4.72430459287232144085620175508, 5.62467019608805364693917132863, 6.30406766751679723499919846732, 7.21306664862750169862317260031, 7.74561054309613471995023118514