| L(s) = 1 | + (−9.79 − 5.65i)2-s + (63.9 + 110. i)4-s + (846. + 488. i)5-s + (1.03e3 + 2.16e3i)7-s − 1.44e3i·8-s + (−5.52e3 − 9.57e3i)10-s + (8.69e3 − 5.01e3i)11-s + 2.83e4·13-s + (2.16e3 − 2.70e4i)14-s + (−8.19e3 + 1.41e4i)16-s + (1.30e5 − 7.55e4i)17-s + (9.01e4 − 1.56e5i)19-s + 1.25e5i·20-s − 1.13e5·22-s + (−1.79e5 − 1.03e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (1.35 + 0.781i)5-s + (0.429 + 0.903i)7-s − 0.353i·8-s + (−0.552 − 0.957i)10-s + (0.593 − 0.342i)11-s + 0.993·13-s + (0.0562 − 0.704i)14-s + (−0.125 + 0.216i)16-s + (1.56 − 0.904i)17-s + (0.691 − 1.19i)19-s + 0.781i·20-s − 0.484·22-s + (−0.639 − 0.369i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0754i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.997 - 0.0754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.557918572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.557918572\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (9.79 + 5.65i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.03e3 - 2.16e3i)T \) |
| good | 5 | \( 1 + (-846. - 488. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-8.69e3 + 5.01e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.83e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.30e5 + 7.55e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.01e4 + 1.56e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.79e5 + 1.03e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.19e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (8.82e5 + 1.52e6i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.01e6 + 1.75e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 3.62e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.49e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.76e6 - 1.59e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (1.01e7 - 5.84e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-6.93e6 + 4.00e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-9.78e6 + 1.69e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-2.39e6 - 4.14e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.83e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.43e7 + 2.49e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (6.04e6 - 1.04e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 5.87e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-8.51e6 - 4.91e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 8.66e7T + 7.83e15T^{2} \) |
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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
−11.52035232430544045320751225889, −10.81759502966584608937710255129, −9.546411107082206515914659512035, −9.071944682993187645933766748216, −7.60880934846348020572994181421, −6.29493156074635963660857114770, −5.41989214792130833099973906648, −3.27068869258566308452973966469, −2.23401340890108857527088472192, −1.06355711957871003851235387155,
1.18420586407644305793891732414, 1.56158960366058816483632373709, 3.85062148590858363139810181710, 5.44028852383324863964884612679, 6.20574165107685929980701263588, 7.67782825799845011116258828759, 8.617038112106548642485719430090, 9.867413067548731436935546721599, 10.23937406046882614576647093355, 11.73748009976523132406509334304
