| 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
−14.99100334792448288009007486701, −13.43297626317000737808448603808, −12.55853845309464343298232797510, −10.85984513763051743478263942174, −9.749467591986847634467438554137, −9.326529339092150373743528541228, −8.559791429054792510784424436311, −4.96962116949953651523910731334, −3.32385649932290051278340749239, −2.36040101450877375421698774263,
0.41693196533607677297104220067, 2.65047184203098597519111372568, 6.14240178954922069767826599218, 7.06680081196063069311201900609, 7.52745166679712360322850367318, 9.133761970405331141389579205724, 9.992378701810980786758878087767, 12.79142141474293086890550673563, 13.60472773757291167164091454824, 14.50080721975777466924049614281
