| L(s) = 1 | − 3.17·2-s + 2.07·4-s + 6.74·5-s + 14.1·7-s + 18.8·8-s − 21.3·10-s + 62.4·11-s − 44.9·14-s − 76.2·16-s + 58.6·17-s − 64.1·19-s + 13.9·20-s − 198.·22-s − 10.9·23-s − 79.5·25-s + 29.3·28-s − 216.·29-s + 38.6·31-s + 91.5·32-s − 186.·34-s + 95.5·35-s − 423.·37-s + 203.·38-s + 126.·40-s − 366.·41-s − 128.·43-s + 129.·44-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.258·4-s + 0.602·5-s + 0.765·7-s + 0.831·8-s − 0.676·10-s + 1.71·11-s − 0.858·14-s − 1.19·16-s + 0.836·17-s − 0.774·19-s + 0.156·20-s − 1.92·22-s − 0.0990·23-s − 0.636·25-s + 0.198·28-s − 1.38·29-s + 0.223·31-s + 0.505·32-s − 0.938·34-s + 0.461·35-s − 1.88·37-s + 0.869·38-s + 0.501·40-s − 1.39·41-s − 0.455·43-s + 0.443·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.17T + 8T^{2} \) |
| 5 | \( 1 - 6.74T + 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 11 | \( 1 - 62.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 58.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 10.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 423.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 366.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 128.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 93.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 131.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 386.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 621.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 865.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 607.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 980.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 907.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−8.807577809816681823021760185729, −8.131623322587866682716402054087, −7.25921819512211120762740661915, −6.46487429112078638706839296131, −5.46609959893299852644635713050, −4.47605292190171023891811954569, −3.55190096265359456369613923541, −1.72239732229836966956343623017, −1.52902080941586266198311217113, 0,
1.52902080941586266198311217113, 1.72239732229836966956343623017, 3.55190096265359456369613923541, 4.47605292190171023891811954569, 5.46609959893299852644635713050, 6.46487429112078638706839296131, 7.25921819512211120762740661915, 8.131623322587866682716402054087, 8.807577809816681823021760185729
