| L(s) = 1 | + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (−25.0 − 18.1i)5-s + (−17.6 + 54.3i)7-s + (−19.7 − 60.8i)8-s − 123.·10-s + (318. + 244. i)11-s + (907. − 659. i)13-s + (70.6 + 217. i)14-s + (−207. − 150. i)16-s + (−1.04e3 − 758. i)17-s + (−609. − 1.87e3i)19-s + (−400. + 290. i)20-s + (1.60e3 + 42.0i)22-s − 4.77e3·23-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.447 − 0.324i)5-s + (−0.136 + 0.419i)7-s + (−0.109 − 0.336i)8-s − 0.390·10-s + (0.793 + 0.608i)11-s + (1.48 − 1.08i)13-s + (0.0963 + 0.296i)14-s + (−0.202 − 0.146i)16-s + (−0.876 − 0.636i)17-s + (−0.387 − 1.19i)19-s + (−0.223 + 0.162i)20-s + (0.706 + 0.0185i)22-s − 1.88·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.727029538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727029538\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-318. - 244. i)T \) |
| good | 5 | \( 1 + (25.0 + 18.1i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (17.6 - 54.3i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-907. + 659. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.04e3 + 758. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (609. + 1.87e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + 4.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (1.72e3 - 5.31e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-11.5 + 8.36i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.33e3 + 1.02e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (684. + 2.10e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 2.20e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (3.74e3 + 1.15e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-6.15e3 + 4.46e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-5.36e3 + 1.65e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.54e4 - 1.12e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 9.17e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (4.31e4 + 3.13e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-8.48e3 + 2.61e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.84e4 + 2.06e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.21e4 - 4.51e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 4.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (4.88e4 - 3.55e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.38804598926544033263108972649, −10.43281253405613769752040510438, −9.191596735302982603300247820714, −8.324515257458853431506393627732, −6.85156752491298931444220730628, −5.81152814470795318733806143253, −4.53062748431665592847574884664, −3.52812384965517200900539712616, −2.02455642761426045220183571372, −0.42521590814500197313491030504,
1.68453738248526718434285033734, 3.74001425982409292763052944509, 4.07463825843656767591451981105, 6.08116718959291922750087251195, 6.51151608594020980785713547633, 7.933903723550706474545815951135, 8.722308314034113111301029543715, 10.13240767834265030940180621156, 11.37711840523876981029849616536, 11.79486047683607722774583388534
