| L(s) = 1 | + (5.34 + 1.64i)2-s + (−3.67 + 9.35i)3-s + (−0.576 − 0.393i)4-s + (96.5 − 14.5i)5-s + (−35.0 + 43.9i)6-s + (41.8 + 122. i)7-s + (−114. − 143. i)8-s + (104. + 96.5i)9-s + (540. + 81.4i)10-s + (−160. + 149. i)11-s + (5.79 − 3.95i)12-s + (241. + 1.05e3i)13-s + (21.3 + 725. i)14-s + (−218. + 956. i)15-s + (−365. − 931. i)16-s + (−132. − 1.77e3i)17-s + ⋯ |
| L(s) = 1 | + (0.945 + 0.291i)2-s + (−0.235 + 0.600i)3-s + (−0.0180 − 0.0122i)4-s + (1.72 − 0.260i)5-s + (−0.397 + 0.498i)6-s + (0.322 + 0.946i)7-s + (−0.630 − 0.790i)8-s + (0.428 + 0.397i)9-s + (1.70 + 0.257i)10-s + (−0.400 + 0.371i)11-s + (0.0116 − 0.00791i)12-s + (0.396 + 1.73i)13-s + (0.0291 + 0.988i)14-s + (−0.250 + 1.09i)15-s + (−0.357 − 0.910i)16-s + (−0.111 − 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.56340 + 1.39926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.56340 + 1.39926i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-41.8 - 122. i)T \) |
| good | 2 | \( 1 + (-5.34 - 1.64i)T + (26.4 + 18.0i)T^{2} \) |
| 3 | \( 1 + (3.67 - 9.35i)T + (-178. - 165. i)T^{2} \) |
| 5 | \( 1 + (-96.5 + 14.5i)T + (2.98e3 - 921. i)T^{2} \) |
| 11 | \( 1 + (160. - 149. i)T + (1.20e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-241. - 1.05e3i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (132. + 1.77e3i)T + (-1.40e6 + 2.11e5i)T^{2} \) |
| 19 | \( 1 + (-548. + 950. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-98.4 + 1.31e3i)T + (-6.36e6 - 9.59e5i)T^{2} \) |
| 29 | \( 1 + (1.27e3 + 616. i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (2.96e3 + 5.12e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-7.07e3 + 4.82e3i)T + (2.53e7 - 6.45e7i)T^{2} \) |
| 41 | \( 1 + (2.83e3 + 3.55e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (3.44e3 - 4.31e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (2.50e4 + 7.71e3i)T + (1.89e8 + 1.29e8i)T^{2} \) |
| 53 | \( 1 + (3.82e3 + 2.60e3i)T + (1.52e8 + 3.89e8i)T^{2} \) |
| 59 | \( 1 + (2.55e4 + 3.84e3i)T + (6.83e8 + 2.10e8i)T^{2} \) |
| 61 | \( 1 + (-2.26e4 + 1.54e4i)T + (3.08e8 - 7.86e8i)T^{2} \) |
| 67 | \( 1 + (-1.68e4 - 2.92e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-4.66e4 + 2.24e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (-4.56e4 + 1.40e4i)T + (1.71e9 - 1.16e9i)T^{2} \) |
| 79 | \( 1 + (5.13e3 - 8.89e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (7.96e3 - 3.49e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (6.21e4 + 5.76e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 - 8.27e4T + 8.58e9T^{2} \) |
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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
−14.54947677128257721470605158196, −13.70682097600664486894002378807, −12.91540332189208910863333397491, −11.39638924966059569026483741553, −9.623318535015851262972710015800, −9.296601649534649630533652360637, −6.61258455931605194565953959362, −5.36354258242412292944865947214, −4.68520500967651836187286570262, −2.14927017094597663717635233316,
1.47211011580157864988285050997, 3.39277851494700113659840889687, 5.37600973573661013004509988614, 6.29612216572995283149079726371, 8.096494799572911788189355867587, 9.946118783758324429700341964203, 10.91318566615703513605458861350, 12.79591637226663693887362577863, 13.09678038964351776594349844194, 14.00596015586001159035784368579
