| L(s) = 1 | + 4.33·2-s + (−9.83 − 17.0i)3-s − 493.·4-s + (215. + 374. i)5-s + (−42.6 − 73.9i)6-s + (2.23e3 − 3.86e3i)7-s − 4.36e3·8-s + (9.64e3 − 1.67e4i)9-s + (936. + 1.62e3i)10-s − 2.74e4·11-s + (4.85e3 + 8.40e3i)12-s + (−4.60e4 + 7.96e4i)13-s + (9.67e3 − 1.67e4i)14-s + (4.24e3 − 7.35e3i)15-s + 2.33e5·16-s + (−2.00e5 + 3.47e5i)17-s + ⋯ |
| L(s) = 1 | + 0.191·2-s + (−0.0701 − 0.121i)3-s − 0.963·4-s + (0.154 + 0.267i)5-s + (−0.0134 − 0.0232i)6-s + (0.351 − 0.608i)7-s − 0.376·8-s + (0.490 − 0.848i)9-s + (0.0296 + 0.0513i)10-s − 0.564·11-s + (0.0675 + 0.116i)12-s + (−0.446 + 0.773i)13-s + (0.0673 − 0.116i)14-s + (0.0216 − 0.0375i)15-s + 0.891·16-s + (−0.583 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.966421 + 0.746021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.966421 + 0.746021i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.88e7 - 1.21e7i)T \) |
| good | 2 | \( 1 - 4.33T + 512T^{2} \) |
| 3 | \( 1 + (9.83 + 17.0i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-215. - 374. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-2.23e3 + 3.86e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + 2.74e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (4.60e4 - 7.96e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (2.00e5 - 3.47e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-3.90e5 - 6.75e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-5.76e5 - 9.97e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (1.36e6 - 2.36e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-2.80e6 - 4.85e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-5.55e6 - 9.61e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 8.96e6T + 3.27e14T^{2} \) |
| 47 | \( 1 + 3.27e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (2.28e7 + 3.95e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 - 9.98e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + (5.24e7 - 9.08e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.37e8 + 2.38e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.39e8 + 2.42e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-8.73e7 + 1.51e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.14e8 - 1.97e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.61e8 - 6.26e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-4.13e8 - 7.16e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.08e9T + 7.60e17T^{2} \) |
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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
−14.18142589956284098280239458476, −13.12978849487364876279427312334, −12.08011619834282240564408652954, −10.45333115901992452839894805282, −9.413613145786792136156442979082, −7.982818162029670136729298448852, −6.48012437248320298614052403504, −4.82823165005702951975533604672, −3.58357475232398006669311803266, −1.27887611637413925467790020094,
0.46571649233901808448794144785, 2.60527728602465310796968382154, 4.70060426086382572939323317229, 5.35574812315613643687053704803, 7.55675794117131983373475256663, 8.846056287411006104141069372859, 9.932010345497387266223515233152, 11.37143140754048305466839173626, 12.91714265196086923798344972890, 13.46780147923348996518987880610
