| L(s) = 1 | − 2.16·2-s − 6.51·3-s − 3.29·4-s − 10.9·5-s + 14.1·6-s − 25.2·7-s + 24.5·8-s + 15.3·9-s + 23.6·10-s + 21.4·12-s + 30.9·13-s + 54.8·14-s + 71.0·15-s − 26.8·16-s + 17·17-s − 33.3·18-s + 145.·19-s + 35.9·20-s + 164.·21-s + 94.4·23-s − 159.·24-s − 5.84·25-s − 67.2·26-s + 75.5·27-s + 83.2·28-s + 24.6·29-s − 154.·30-s + ⋯ |
| L(s) = 1 | − 0.766·2-s − 1.25·3-s − 0.411·4-s − 0.976·5-s + 0.961·6-s − 1.36·7-s + 1.08·8-s + 0.569·9-s + 0.748·10-s + 0.515·12-s + 0.661·13-s + 1.04·14-s + 1.22·15-s − 0.418·16-s + 0.242·17-s − 0.437·18-s + 1.75·19-s + 0.401·20-s + 1.71·21-s + 0.856·23-s − 1.35·24-s − 0.0467·25-s − 0.507·26-s + 0.538·27-s + 0.562·28-s + 0.157·29-s − 0.938·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4431722483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4431722483\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 2.16T + 8T^{2} \) |
| 3 | \( 1 + 6.51T + 27T^{2} \) |
| 5 | \( 1 + 10.9T + 125T^{2} \) |
| 7 | \( 1 + 25.2T + 343T^{2} \) |
| 13 | \( 1 - 30.9T + 2.19e3T^{2} \) |
| 19 | \( 1 - 145.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 94.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 24.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 43.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 446.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 489.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 478.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 630.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 492.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 139.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 626.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 74.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 103.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 407.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 497.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3T + 9.12e5T^{2} \) |
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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
−8.987749867240052281805833620642, −7.85212297231601353996952694073, −7.36975429291603457863311951889, −6.42601240719090000127855207780, −5.68371754991906249848349037833, −4.81602654143862896178606558468, −3.86765646561126062465208051650, −3.06535240948782712705176394688, −1.02804262973590517597039781740, −0.48999234964827516054797111808,
0.48999234964827516054797111808, 1.02804262973590517597039781740, 3.06535240948782712705176394688, 3.86765646561126062465208051650, 4.81602654143862896178606558468, 5.68371754991906249848349037833, 6.42601240719090000127855207780, 7.36975429291603457863311951889, 7.85212297231601353996952694073, 8.987749867240052281805833620642
