| L(s) = 1 | + (−6.07 + 12.6i)2-s + (28.9 − 42.5i)3-s + (−82.3 − 103. i)4-s + (88.1 + 95.0i)5-s + (360. + 623. i)6-s + (−445. − 257. i)7-s + (930. − 212. i)8-s + (−700. − 1.78e3i)9-s + (−1.73e3 + 535. i)10-s + (−442. + 555. i)11-s + (−6.77e3 + 507. i)12-s + (−2.67e3 − 825. i)13-s + (5.95e3 − 4.06e3i)14-s + (6.59e3 − 993. i)15-s + (−1.09e3 + 4.78e3i)16-s + (−1.32e3 − 1.22e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.759 + 1.57i)2-s + (1.07 − 1.57i)3-s + (−1.28 − 1.61i)4-s + (0.705 + 0.760i)5-s + (1.66 + 2.88i)6-s + (−1.30 − 0.750i)7-s + (1.81 − 0.414i)8-s + (−0.961 − 2.44i)9-s + (−1.73 + 0.535i)10-s + (−0.332 + 0.417i)11-s + (−3.92 + 0.293i)12-s + (−1.21 − 0.375i)13-s + (2.17 − 1.48i)14-s + (1.95 − 0.294i)15-s + (−0.266 + 1.16i)16-s + (−0.269 − 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.628899 - 0.549423i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.628899 - 0.549423i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.03e3 + 7.94e4i)T \) |
| good | 2 | \( 1 + (6.07 - 12.6i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-28.9 + 42.5i)T + (-266. - 678. i)T^{2} \) |
| 5 | \( 1 + (-88.1 - 95.0i)T + (-1.16e3 + 1.55e4i)T^{2} \) |
| 7 | \( 1 + (445. + 257. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (442. - 555. i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (2.67e3 + 825. i)T + (3.98e6 + 2.71e6i)T^{2} \) |
| 17 | \( 1 + (1.32e3 + 1.22e3i)T + (1.80e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (1.75e3 + 687. i)T + (3.44e7 + 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-1.59e4 - 2.40e3i)T + (1.41e8 + 4.36e7i)T^{2} \) |
| 29 | \( 1 + (8.81e3 + 1.29e4i)T + (-2.17e8 + 5.53e8i)T^{2} \) |
| 31 | \( 1 + (1.65e3 + 2.20e4i)T + (-8.77e8 + 1.32e8i)T^{2} \) |
| 37 | \( 1 + (-2.52e4 + 1.45e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (1.92e4 + 9.28e3i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-9.02e4 - 1.13e5i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (1.17e5 - 3.62e4i)T + (1.83e10 - 1.24e10i)T^{2} \) |
| 59 | \( 1 + (2.54e4 - 1.11e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.23e5 - 9.23e3i)T + (5.09e10 + 7.67e9i)T^{2} \) |
| 67 | \( 1 + (1.92e4 - 4.90e4i)T + (-6.63e10 - 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-6.41e4 - 4.25e5i)T + (-1.22e11 + 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-3.08e4 + 9.99e4i)T + (-1.25e11 - 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-2.88e5 + 4.99e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (2.09e5 + 1.42e5i)T + (1.19e11 + 3.04e11i)T^{2} \) |
| 89 | \( 1 + (-3.95e5 + 5.80e5i)T + (-1.81e11 - 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-8.98e5 + 1.12e6i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−14.50080721975777466924049614281, −13.60472773757291167164091454824, −12.79142141474293086890550673563, −9.992378701810980786758878087767, −9.133761970405331141389579205724, −7.52745166679712360322850367318, −7.06680081196063069311201900609, −6.14240178954922069767826599218, −2.65047184203098597519111372568, −0.41693196533607677297104220067,
2.36040101450877375421698774263, 3.32385649932290051278340749239, 4.96962116949953651523910731334, 8.559791429054792510784424436311, 9.326529339092150373743528541228, 9.749467591986847634467438554137, 10.85984513763051743478263942174, 12.55853845309464343298232797510, 13.43297626317000737808448603808, 14.99100334792448288009007486701
