| L(s) = 1 | − 3.33·3-s − 0.932·5-s − 3.45·7-s + 8.09·9-s + 1.85·11-s + 3.10·15-s + 0.218·17-s + 5.10·19-s + 11.5·21-s − 7.49·23-s − 4.13·25-s − 16.9·27-s − 3.19·29-s − 7.91·31-s − 6.19·33-s + 3.22·35-s − 4.26·37-s − 3.27·41-s + 4.58·43-s − 7.55·45-s − 1.30·47-s + 4.95·49-s − 0.727·51-s − 2.68·53-s − 1.73·55-s − 17.0·57-s + 9.23·59-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 0.417·5-s − 1.30·7-s + 2.69·9-s + 0.560·11-s + 0.802·15-s + 0.0529·17-s + 1.17·19-s + 2.51·21-s − 1.56·23-s − 0.826·25-s − 3.26·27-s − 0.593·29-s − 1.42·31-s − 1.07·33-s + 0.545·35-s − 0.701·37-s − 0.510·41-s + 0.699·43-s − 1.12·45-s − 0.191·47-s + 0.707·49-s − 0.101·51-s − 0.369·53-s − 0.233·55-s − 2.25·57-s + 1.20·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3743207639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3743207639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3.33T + 3T^{2} \) |
| 5 | \( 1 + 0.932T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 17 | \( 1 - 0.218T + 17T^{2} \) |
| 19 | \( 1 - 5.10T + 19T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 + 7.91T + 31T^{2} \) |
| 37 | \( 1 + 4.26T + 37T^{2} \) |
| 41 | \( 1 + 3.27T + 41T^{2} \) |
| 43 | \( 1 - 4.58T + 43T^{2} \) |
| 47 | \( 1 + 1.30T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + 4.43T + 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 - 3.28T + 73T^{2} \) |
| 79 | \( 1 + 7.33T + 79T^{2} \) |
| 83 | \( 1 - 3.58T + 83T^{2} \) |
| 89 | \( 1 - 5.07T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−9.134485689142110133124603630178, −7.69693110332998385789667527447, −7.14725959439029762244628315840, −6.37033319998848613884659009480, −5.84121451593483583917027359552, −5.18178379057667395719386365983, −4.03814686770283145865128472113, −3.55290001962095798739099436142, −1.74898532609017671213728075936, −0.41826711553313859668215894465,
0.41826711553313859668215894465, 1.74898532609017671213728075936, 3.55290001962095798739099436142, 4.03814686770283145865128472113, 5.18178379057667395719386365983, 5.84121451593483583917027359552, 6.37033319998848613884659009480, 7.14725959439029762244628315840, 7.69693110332998385789667527447, 9.134485689142110133124603630178
