| L(s) = 1 | + (−4.32 − 1.69i)2-s + (5.16 + 0.591i)3-s + (9.94 + 9.22i)4-s + (7.56 + 5.15i)5-s + (−21.3 − 11.3i)6-s + (16.1 + 9.04i)7-s + (−11.2 − 23.3i)8-s + (26.3 + 6.10i)9-s + (−23.9 − 35.1i)10-s + (4.18 + 27.7i)11-s + (45.8 + 53.5i)12-s + (−55.3 + 44.1i)13-s + (−54.5 − 66.5i)14-s + (35.9 + 31.0i)15-s + (0.864 + 11.5i)16-s + (−76.7 − 23.6i)17-s + ⋯ |
| L(s) = 1 | + (−1.52 − 0.599i)2-s + (0.993 + 0.113i)3-s + (1.24 + 1.15i)4-s + (0.676 + 0.461i)5-s + (−1.45 − 0.769i)6-s + (0.872 + 0.488i)7-s + (−0.495 − 1.02i)8-s + (0.974 + 0.226i)9-s + (−0.757 − 1.11i)10-s + (0.114 + 0.761i)11-s + (1.10 + 1.28i)12-s + (−1.17 + 0.940i)13-s + (−1.04 − 1.27i)14-s + (0.619 + 0.535i)15-s + (0.0135 + 0.180i)16-s + (−1.09 − 0.337i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.16262 + 0.433381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16262 + 0.433381i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.16 - 0.591i)T \) |
| 7 | \( 1 + (-16.1 - 9.04i)T \) |
| good | 2 | \( 1 + (4.32 + 1.69i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-7.56 - 5.15i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (-4.18 - 27.7i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (55.3 - 44.1i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (76.7 + 23.6i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (73.4 - 42.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-41.5 - 134. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-275. + 62.9i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (99.1 + 57.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-172. + 159. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (205. - 98.8i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-224. - 108. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-145. + 369. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-14.1 + 15.2i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-237. + 161. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (261. + 281. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (124. - 215. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (240. + 54.9i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-560. + 219. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-311. - 540. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-220. + 276. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-397. - 59.9i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 393. iT - 9.12e5T^{2} \) |
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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
−12.38774574028896690884714668597, −11.39699987909302678869335554176, −10.27107088008277570732833358728, −9.549794011163433763050900467389, −8.838164480020586353699910761747, −7.79094746645498951989130889588, −6.84276825477553434896285293069, −4.58552701278160019051411875055, −2.39805518507144873521545388873, −1.93265929817921408736231161996,
0.905165710993478108057039059930, 2.34945214953608590084307194092, 4.69366474056251379935760756336, 6.47789081235992279843333503704, 7.53436726222598093933329956016, 8.515494983793057734889942422478, 8.945370447107719869221844287538, 10.18096997447814853888902500446, 10.81101841570427470989420455758, 12.64684606788976865295875773604
