L(s) = 1 | + 1.75·2-s − 0.900·3-s + 1.07·4-s + 1.37·5-s − 1.58·6-s − 1.58·7-s − 1.61·8-s − 2.18·9-s + 2.41·10-s + 11-s − 0.970·12-s − 0.360·13-s − 2.77·14-s − 1.24·15-s − 4.99·16-s − 1.83·17-s − 3.83·18-s − 5.46·19-s + 1.48·20-s + 1.42·21-s + 1.75·22-s + 4.12·23-s + 1.45·24-s − 3.10·25-s − 0.632·26-s + 4.67·27-s − 1.70·28-s + ⋯ |
L(s) = 1 | + 1.24·2-s − 0.520·3-s + 0.538·4-s + 0.616·5-s − 0.645·6-s − 0.598·7-s − 0.572·8-s − 0.729·9-s + 0.764·10-s + 0.301·11-s − 0.280·12-s − 0.0999·13-s − 0.742·14-s − 0.320·15-s − 1.24·16-s − 0.445·17-s − 0.904·18-s − 1.25·19-s + 0.331·20-s + 0.311·21-s + 0.374·22-s + 0.859·23-s + 0.297·24-s − 0.620·25-s − 0.123·26-s + 0.899·27-s − 0.322·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 3 | \( 1 + 0.900T + 3T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 13 | \( 1 + 0.360T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 6.72T + 41T^{2} \) |
| 43 | \( 1 + 5.05T + 43T^{2} \) |
| 47 | \( 1 - 8.96T + 47T^{2} \) |
| 53 | \( 1 - 9.98T + 53T^{2} \) |
| 59 | \( 1 - 0.723T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 + 5.91T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 - 0.471T + 73T^{2} \) |
| 79 | \( 1 - 2.77T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 + 0.369T + 97T^{2} \) |
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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
−9.008197251368622843623315425866, −8.596802204418705676779201746958, −6.92628983931560384708374259256, −6.46138080478724075792704457415, −5.63891595711242088532975306872, −5.10813696161574011567569187837, −4.04805229915058678264088449202, −3.15646602330540055760235670177, −2.11409412679933626901114619149, 0,
2.11409412679933626901114619149, 3.15646602330540055760235670177, 4.04805229915058678264088449202, 5.10813696161574011567569187837, 5.63891595711242088532975306872, 6.46138080478724075792704457415, 6.92628983931560384708374259256, 8.596802204418705676779201746958, 9.008197251368622843623315425866