| L(s) = 1 | + (1.38 − 7.87i)2-s + (45.5 + 38.2i)3-s + (−60.1 − 21.8i)4-s + (−199. + 72.5i)5-s + (364. − 305. i)6-s + (458. + 793. i)7-s + (−256 + 443. i)8-s + (234. + 1.33e3i)9-s + (294. + 1.67e3i)10-s + (−2.15e3 + 3.73e3i)11-s + (−1.90e3 − 3.29e3i)12-s + (−5.42e3 + 4.55e3i)13-s + (6.89e3 − 2.50e3i)14-s + (−1.18e4 − 4.31e3i)15-s + (3.13e3 + 2.63e3i)16-s + (−918. + 5.21e3i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.974 + 0.817i)3-s + (−0.469 − 0.171i)4-s + (−0.712 + 0.259i)5-s + (0.688 − 0.578i)6-s + (0.505 + 0.874i)7-s + (−0.176 + 0.306i)8-s + (0.107 + 0.608i)9-s + (0.0931 + 0.528i)10-s + (−0.488 + 0.846i)11-s + (−0.317 − 0.550i)12-s + (−0.684 + 0.574i)13-s + (0.671 − 0.244i)14-s + (−0.906 − 0.329i)15-s + (0.191 + 0.160i)16-s + (−0.0453 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.50140 + 1.14693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50140 + 1.14693i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 7.87i)T \) |
| 19 | \( 1 + (-2.13e4 - 2.08e4i)T \) |
| good | 3 | \( 1 + (-45.5 - 38.2i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (199. - 72.5i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-458. - 793. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.15e3 - 3.73e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (5.42e3 - 4.55e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (918. - 5.21e3i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-8.60e4 - 3.13e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (2.62e4 + 1.48e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-6.80e3 - 1.17e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.63e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (6.58e4 + 5.52e4i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-7.01e5 + 2.55e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-2.23e4 - 1.26e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (9.12e5 + 3.31e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-1.88e5 + 1.07e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (2.05e5 + 7.47e4i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-2.65e5 - 1.50e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-1.84e6 + 6.71e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-7.33e5 - 6.15e5i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-5.86e6 - 4.92e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (1.58e6 + 2.73e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.33e6 + 1.11e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (2.21e6 - 1.25e7i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−15.07175134002913619383080317417, −14.07147144095659759800446484518, −12.42823646774466759564227116339, −11.41158982067521147604577886200, −9.944503885760337338447861315253, −8.989248272032770016101982163240, −7.66970993040720865899080197075, −5.00365407621635855707122323036, −3.64071571187592667234466493891, −2.25747690005932184836976438909,
0.73218784792908196196155737058, 3.10662928448223685299411325685, 4.94019645236454109300568601709, 7.17144996375567977712009919644, 7.83136551577175676439699672701, 8.882487238499735279025891536361, 10.89051113423095918399287129835, 12.57542787741275890354541749773, 13.59262603468404681389757237249, 14.37474335621371956718189635660
