| L(s) = 1 | + (13.0 − 9.46i)2-s + (−15.7 + 48.4i)3-s + (40.6 − 124. i)4-s + (−105. + 258. i)5-s + (253. + 780. i)6-s + 1.54e3·7-s + (−16.9 − 52.2i)8-s + (−333. − 242. i)9-s + (1.07e3 + 4.37e3i)10-s + (−3.27e3 + 2.37e3i)11-s + (5.42e3 + 3.93e3i)12-s + (−2.40e3 − 1.74e3i)13-s + (2.01e4 − 1.46e4i)14-s + (−1.08e4 − 9.20e3i)15-s + (1.28e4 + 9.36e3i)16-s + (−8.75e3 − 2.69e4i)17-s + ⋯ |
| L(s) = 1 | + (1.15 − 0.836i)2-s + (−0.336 + 1.03i)3-s + (0.317 − 0.976i)4-s + (−0.378 + 0.925i)5-s + (0.479 + 1.47i)6-s + 1.70·7-s + (−0.0117 − 0.0360i)8-s + (−0.152 − 0.110i)9-s + (0.339 + 1.38i)10-s + (−0.741 + 0.538i)11-s + (0.905 + 0.657i)12-s + (−0.303 − 0.220i)13-s + (1.96 − 1.42i)14-s + (−0.832 − 0.703i)15-s + (0.786 + 0.571i)16-s + (−0.432 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.67199 + 0.836580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.67199 + 0.836580i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (105. - 258. i)T \) |
| good | 2 | \( 1 + (-13.0 + 9.46i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (15.7 - 48.4i)T + (-1.76e3 - 1.28e3i)T^{2} \) |
| 7 | \( 1 - 1.54e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + (3.27e3 - 2.37e3i)T + (6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (2.40e3 + 1.74e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (8.75e3 + 2.69e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-4.46e3 - 1.37e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-3.76e4 + 2.73e4i)T + (1.05e9 - 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-6.89e4 + 2.12e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-1.39e4 - 4.30e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.45e5 - 1.78e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (2.87e5 + 2.09e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 - 3.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.39e5 + 7.36e5i)T + (-4.09e11 - 2.97e11i)T^{2} \) |
| 53 | \( 1 + (3.54e5 - 1.09e6i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (4.93e5 + 3.58e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-1.22e6 + 8.90e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (2.26e5 + 6.95e5i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (-8.24e5 + 2.53e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (3.88e6 - 2.82e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-9.03e5 + 2.77e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (7.49e5 + 2.30e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 + (9.52e6 - 6.92e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (3.05e6 - 9.40e6i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−15.52995610254316167946535946170, −14.79653461870401311893712032416, −13.73466485770638551217853842753, −11.89197836827053521238046045746, −11.10127779263912567752579420730, −10.24424365700701046928325889507, −7.75437466793935419285451265017, −5.13309792061619228678892057345, −4.33268334110971283452573839966, −2.48164537160539196700327208818,
1.26449511223174595941420086319, 4.49778430836215557470729373435, 5.60790507523943486158478671906, 7.30211119912167091058180981371, 8.335055712417278344508575591758, 11.25401290559102611396700167077, 12.51716560882060287000356185552, 13.28338668184885496401885475528, 14.54317639900016323674585854148, 15.64364718124572960070788681843
