| L(s) = 1 | + (−0.534 + 1.04i)2-s + (5.70 − 0.902i)3-s + (1.53 + 2.11i)4-s + (−2.10 + 6.46i)6-s + (−1.94 − 0.308i)7-s + (−7.69 + 1.21i)8-s + (23.1 − 7.51i)9-s + (−9.40 + 5.71i)11-s + (10.6 + 10.6i)12-s + (14.8 + 7.55i)13-s + (1.36 − 1.88i)14-s + (−0.394 + 1.21i)16-s + (4.31 − 2.20i)17-s + (−4.48 + 28.2i)18-s + (−8.23 + 11.3i)19-s + ⋯ |
| L(s) = 1 | + (−0.267 + 0.524i)2-s + (1.90 − 0.300i)3-s + (0.383 + 0.528i)4-s + (−0.350 + 1.07i)6-s + (−0.278 − 0.0441i)7-s + (−0.961 + 0.152i)8-s + (2.56 − 0.834i)9-s + (−0.854 + 0.519i)11-s + (0.888 + 0.888i)12-s + (1.14 + 0.581i)13-s + (0.0976 − 0.134i)14-s + (−0.0246 + 0.0758i)16-s + (0.254 − 0.129i)17-s + (−0.248 + 1.57i)18-s + (−0.433 + 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 - 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.579 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.43811 + 1.25884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43811 + 1.25884i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (9.40 - 5.71i)T \) |
| good | 2 | \( 1 + (0.534 - 1.04i)T + (-2.35 - 3.23i)T^{2} \) |
| 3 | \( 1 + (-5.70 + 0.902i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (1.94 + 0.308i)T + (46.6 + 15.1i)T^{2} \) |
| 13 | \( 1 + (-14.8 - 7.55i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-4.31 + 2.20i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (8.23 - 11.3i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-20.5 + 20.5i)T - 529iT^{2} \) |
| 29 | \( 1 + (25.5 + 35.2i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (3.79 + 11.6i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-0.192 - 0.0304i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (35.3 + 25.7i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-19.6 + 19.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.69 - 48.5i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (5.00 + 2.54i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-1.39 - 1.92i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (36.3 - 111. i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (50.5 + 50.5i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + (0.478 - 1.47i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-14.1 + 89.5i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-47.9 + 15.5i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-6.34 - 12.4i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + 80.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (49.3 - 96.8i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−12.13157290940805186198906810624, −10.63101265777318148193338866612, −9.446255370794525711279713817079, −8.722094679602851774078936707799, −7.953251839067771219736122263379, −7.28099767703188805857280392105, −6.28365437973100125091688479730, −4.12109422367990884635914554321, −3.09848325588507131216576722600, −2.03753672687634000146536513172,
1.50889963967801575306859832552, 2.89014370146966013628084110712, 3.48431387972646322940417310090, 5.28866519842802718561617754914, 6.81153104953239476688097142794, 7.987242241680624998898995887748, 8.828248193030338085746362686185, 9.521288694076631051408554347383, 10.49148708582440297649021627915, 11.08912038695047128374408729673
