| L(s) = 1 | + (3.61 + 1.41i)2-s + (5.01 + 1.36i)3-s + (5.18 + 4.80i)4-s + (14.1 + 9.62i)5-s + (16.1 + 12.0i)6-s + (−17.7 − 5.35i)7-s + (−1.57 − 3.26i)8-s + (23.2 + 13.6i)9-s + (37.3 + 54.7i)10-s + (−4.02 − 26.6i)11-s + (19.4 + 31.1i)12-s + (−20.2 + 16.1i)13-s + (−56.4 − 44.4i)14-s + (57.6 + 67.5i)15-s + (−5.27 − 70.3i)16-s + (−25.6 − 7.91i)17-s + ⋯ |
| L(s) = 1 | + (1.27 + 0.501i)2-s + (0.964 + 0.262i)3-s + (0.647 + 0.600i)4-s + (1.26 + 0.860i)5-s + (1.10 + 0.818i)6-s + (−0.957 − 0.288i)7-s + (−0.0694 − 0.144i)8-s + (0.862 + 0.506i)9-s + (1.18 + 1.73i)10-s + (−0.110 − 0.731i)11-s + (0.467 + 0.749i)12-s + (−0.432 + 0.344i)13-s + (−1.07 − 0.849i)14-s + (0.992 + 1.16i)15-s + (−0.0824 − 1.09i)16-s + (−0.366 − 0.112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.91866 + 2.13291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.91866 + 2.13291i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.01 - 1.36i)T \) |
| 7 | \( 1 + (17.7 + 5.35i)T \) |
| good | 2 | \( 1 + (-3.61 - 1.41i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-14.1 - 9.62i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (4.02 + 26.6i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (20.2 - 16.1i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (25.6 + 7.91i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (39.0 - 22.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (35.6 + 115. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-172. + 39.4i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-219. - 126. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (256. - 238. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (232. - 111. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (279. + 134. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-149. + 380. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (225. - 243. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-656. + 447. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (394. + 425. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (245. - 425. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-571. - 130. i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-261. + 102. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (437. + 758. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (258. - 324. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-163. - 24.6i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 42.3iT - 9.12e5T^{2} \) |
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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
−13.39737820657650577134511010807, −12.24393397925574298358595502081, −10.32939131516034820371784519035, −9.917741425693312333131056126315, −8.576828588664779910649783520288, −6.75988608145745788531801228241, −6.42130742092880237468179748339, −4.88667439141981678795224646500, −3.46012313158858579591634699123, −2.54493061197940370374741607483,
1.89684713343795482835777911952, 2.89971698571645893571216924014, 4.38211926824938562493708902804, 5.55644267624114835425691302974, 6.70605233303792098048488956235, 8.456054733702799236758005720316, 9.445226993276389923939202963726, 10.20366358223441487497070425742, 12.14143927503858465756964686896, 12.69113215097644671192113207395
