L(s) = 1 | + (5.00 − 6.23i)2-s + (−11.0 − 11.0i)3-s + (−13.8 − 62.4i)4-s + (45.7 + 45.7i)5-s + (−123. + 13.5i)6-s − 565.·7-s + (−458. − 226. i)8-s + 242. i·9-s + (514. − 56.1i)10-s + (1.29e3 − 1.29e3i)11-s + (−536. + 841. i)12-s + (−2.88e3 + 2.88e3i)13-s + (−2.83e3 + 3.52e3i)14-s − 1.00e3i·15-s + (−3.71e3 + 1.72e3i)16-s − 662.·17-s + ⋯ |
L(s) = 1 | + (0.626 − 0.779i)2-s + (−0.408 − 0.408i)3-s + (−0.215 − 0.976i)4-s + (0.366 + 0.366i)5-s + (−0.573 + 0.0626i)6-s − 1.64·7-s + (−0.896 − 0.443i)8-s + 0.333i·9-s + (0.514 − 0.0561i)10-s + (0.970 − 0.970i)11-s + (−0.310 + 0.486i)12-s + (−1.31 + 1.31i)13-s + (−1.03 + 1.28i)14-s − 0.299i·15-s + (−0.906 + 0.421i)16-s − 0.134·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.221445 + 0.568215i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.221445 + 0.568215i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.00 + 6.23i)T \) |
| 3 | \( 1 + (11.0 + 11.0i)T \) |
good | 5 | \( 1 + (-45.7 - 45.7i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 565.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-1.29e3 + 1.29e3i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (2.88e3 - 2.88e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 662.T + 2.41e7T^{2} \) |
| 19 | \( 1 + (-1.94e3 - 1.94e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 + 1.58e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-1.15e4 + 1.15e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 3.90e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (5.13e4 + 5.13e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 6.65e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-6.89e4 + 6.89e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 1.07e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.08e5 + 1.08e5i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (6.78e4 - 6.78e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (9.92e4 - 9.92e4i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (1.05e5 + 1.05e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 2.66e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 2.67e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 5.57e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (2.36e5 + 2.36e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 4.19e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.93e5T + 8.32e11T^{2} \) |
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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
−13.66537675722227090979665208047, −12.37324205236260737748423478470, −11.66884901225104666528682092472, −10.13882924466567058679458246178, −9.323176084512378573588766335751, −6.70868182793713874390853418268, −5.98190454098295953522679813624, −3.92261942234703074213802808388, −2.31665537858873895197827886251, −0.22077824399636692929110079186,
3.21571315000693033041433674531, 4.82877984041839360554075142181, 6.12746822980372079863285789165, 7.24614899316044648147606730820, 9.204020468000351911455609325790, 10.05852785591368713867787683324, 12.29876584707835994675276951738, 12.62402509510520481148425948339, 14.05953870163335134148838032349, 15.31543664126766974634499075426