L(s) = 1 | − 0.788·3-s − 3.31·5-s + 3.09·7-s − 2.37·9-s − 1.54·11-s + 4.10·13-s + 2.61·15-s − 5.79·17-s − 0.478·19-s − 2.43·21-s − 4.76·23-s + 6.00·25-s + 4.24·27-s + 4.75·29-s + 7.29·31-s + 1.21·33-s − 10.2·35-s − 2.26·37-s − 3.23·39-s + 6.44·41-s − 1.23·43-s + 7.88·45-s + 3.04·47-s + 2.54·49-s + 4.56·51-s + 1.22·53-s + 5.12·55-s + ⋯ |
L(s) = 1 | − 0.455·3-s − 1.48·5-s + 1.16·7-s − 0.792·9-s − 0.465·11-s + 1.13·13-s + 0.675·15-s − 1.40·17-s − 0.109·19-s − 0.531·21-s − 0.993·23-s + 1.20·25-s + 0.816·27-s + 0.882·29-s + 1.31·31-s + 0.212·33-s − 1.73·35-s − 0.372·37-s − 0.518·39-s + 1.00·41-s − 0.188·43-s + 1.17·45-s + 0.443·47-s + 0.364·49-s + 0.639·51-s + 0.167·53-s + 0.691·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
good | 3 | \( 1 + 0.788T + 3T^{2} \) |
| 5 | \( 1 + 3.31T + 5T^{2} \) |
| 7 | \( 1 - 3.09T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 + 0.478T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 - 4.75T + 29T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 6.44T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 - 1.22T + 53T^{2} \) |
| 59 | \( 1 + 5.53T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 6.29T + 71T^{2} \) |
| 73 | \( 1 - 4.93T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 7.90T + 97T^{2} \) |
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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
−7.65271934096875671085909580936, −6.65672825183801913400425705625, −6.12043258455796348900577776033, −5.22897872494234658053146635065, −4.48562500608504281691953510451, −4.12364967828010701393207493310, −3.12559098147152001106702338193, −2.23604921422499907012709059803, −1.00120381709227408680207376874, 0,
1.00120381709227408680207376874, 2.23604921422499907012709059803, 3.12559098147152001106702338193, 4.12364967828010701393207493310, 4.48562500608504281691953510451, 5.22897872494234658053146635065, 6.12043258455796348900577776033, 6.65672825183801913400425705625, 7.65271934096875671085909580936