| L(s) = 1 | + (−5.98 − 7.50i)2-s + (−64.3 − 9.70i)3-s + (93.4 − 409. i)4-s + (−1.64e3 − 1.12e3i)5-s + (312. + 541. i)6-s + (−2.49e3 + 4.32e3i)7-s + (−8.05e3 + 3.88e3i)8-s + (−1.47e4 − 4.55e3i)9-s + (1.42e3 + 1.90e4i)10-s + (−4.49e3 − 1.96e4i)11-s + (−9.98e3 + 2.54e4i)12-s + (5.96e3 − 7.95e4i)13-s + (4.74e4 − 7.14e3i)14-s + (9.49e4 + 8.81e4i)15-s + (−1.16e5 − 5.60e4i)16-s + (1.56e5 − 1.06e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.264 − 0.331i)2-s + (−0.458 − 0.0691i)3-s + (0.182 − 0.799i)4-s + (−1.17 − 0.802i)5-s + (0.0983 + 0.170i)6-s + (−0.393 + 0.681i)7-s + (−0.695 + 0.334i)8-s + (−0.749 − 0.231i)9-s + (0.0451 + 0.602i)10-s + (−0.0924 − 0.405i)11-s + (−0.138 + 0.354i)12-s + (0.0578 − 0.772i)13-s + (0.330 − 0.0497i)14-s + (0.484 + 0.449i)15-s + (−0.443 − 0.213i)16-s + (0.454 − 0.309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.109616 + 0.0499431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.109616 + 0.0499431i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.78e7 + 1.35e7i)T \) |
| good | 2 | \( 1 + (5.98 + 7.50i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (64.3 + 9.70i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (1.64e3 + 1.12e3i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (2.49e3 - 4.32e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (4.49e3 + 1.96e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (-5.96e3 + 7.95e4i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (-1.56e5 + 1.06e5i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (2.48e4 - 7.66e3i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (4.94e5 - 4.58e5i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (-6.85e6 + 1.03e6i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (1.30e6 - 3.31e6i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (7.80e6 + 1.35e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-1.93e7 - 2.42e7i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (3.43e6 - 1.50e7i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (-1.06e6 - 1.42e7i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (1.25e8 + 6.02e7i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (6.11e7 + 1.55e8i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (3.10e7 - 9.57e6i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (-6.74e7 - 6.26e7i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (-1.43e6 + 1.90e7i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (-6.87e7 + 1.19e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (7.94e8 + 1.19e8i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (-7.74e8 - 1.16e8i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (-2.08e8 - 9.11e8i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−14.26369224947197871395046232508, −12.43229207866913790938984446687, −11.80016066240775871906491102684, −10.71685824241982532482430691152, −9.201488054086238805386087454089, −8.141836277766138776392579522984, −6.15731453148916051438304276625, −5.08725366957298994249401206776, −3.03214787207718867582470551907, −0.865450817760834489450454247469,
0.06719383581584676455309444994, 2.98105233950570744140618895541, 4.19615638431852678114920478715, 6.48153458344673486143384361089, 7.40843762183403619504283850782, 8.500442960551814934228884703074, 10.37954057708886111275389735994, 11.53432127340347626028165723952, 12.27448808292385878099144023431, 13.93415342993879521385447428866
