| L(s) = 1 | + 4·2-s + 16·4-s − 94.9·5-s + 64·8-s − 379.·10-s − 476·11-s − 963.·13-s + 256·16-s − 895.·17-s − 637.·19-s − 1.51e3·20-s − 1.90e3·22-s − 3.69e3·23-s + 5.89e3·25-s − 3.85e3·26-s − 1.39e3·29-s − 1.92e3·31-s + 1.02e3·32-s − 3.58e3·34-s + 1.20e4·37-s − 2.55e3·38-s − 6.07e3·40-s + 1.52e4·41-s + 9.72e3·43-s − 7.61e3·44-s − 1.47e4·46-s − 2.92e4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.69·5-s + 0.353·8-s − 1.20·10-s − 1.18·11-s − 1.58·13-s + 0.250·16-s − 0.751·17-s − 0.405·19-s − 0.849·20-s − 0.838·22-s − 1.45·23-s + 1.88·25-s − 1.11·26-s − 0.307·29-s − 0.359·31-s + 0.176·32-s − 0.531·34-s + 1.45·37-s − 0.286·38-s − 0.600·40-s + 1.41·41-s + 0.801·43-s − 0.593·44-s − 1.03·46-s − 1.93·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6386628270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6386628270\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 94.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 476T + 1.61e5T^{2} \) |
| 13 | \( 1 + 963.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 895.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 637.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.92e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.31e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.29e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.97e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.54e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.77e4T + 8.58e9T^{2} \) |
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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
−9.448910572047226331441477524930, −8.073113077975309939043345265879, −7.76389544854051304723854043807, −6.97556720861362857795028986732, −5.79215773420580455247651187747, −4.63058348646369431673294314824, −4.27003863456086106582371012636, −3.06413154817449539951363824565, −2.22165086031048676654528569414, −0.29701194573982210511927905876,
0.29701194573982210511927905876, 2.22165086031048676654528569414, 3.06413154817449539951363824565, 4.27003863456086106582371012636, 4.63058348646369431673294314824, 5.79215773420580455247651187747, 6.97556720861362857795028986732, 7.76389544854051304723854043807, 8.073113077975309939043345265879, 9.448910572047226331441477524930
