| L(s) = 1 | + (−0.397 − 0.824i)2-s + (14.6 − 30.5i)3-s + (159. − 199. i)4-s + (1.12e3 + 257. i)5-s − 30.9·6-s − 4.07e3i·7-s + (−456. − 104. i)8-s + (3.37e3 + 4.23e3i)9-s + (−236. − 1.03e3i)10-s + (6.05e3 + 7.59e3i)11-s + (−3.74e3 − 7.78e3i)12-s + (−9.96e3 + 4.36e4i)13-s + (−3.35e3 + 1.61e3i)14-s + (2.44e4 − 3.06e4i)15-s + (−1.44e4 − 6.32e4i)16-s + (−1.22e4 − 5.37e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0248 − 0.0515i)2-s + (0.181 − 0.376i)3-s + (0.621 − 0.779i)4-s + (1.80 + 0.412i)5-s − 0.0239·6-s − 1.69i·7-s + (−0.111 − 0.0254i)8-s + (0.514 + 0.645i)9-s + (−0.0236 − 0.103i)10-s + (0.413 + 0.518i)11-s + (−0.180 − 0.375i)12-s + (−0.348 + 1.52i)13-s + (−0.0873 + 0.0420i)14-s + (0.483 − 0.605i)15-s + (−0.220 − 0.965i)16-s + (−0.146 − 0.643i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.71768 - 1.53366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.71768 - 1.53366i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.00e6 + 1.63e6i)T \) |
| good | 2 | \( 1 + (0.397 + 0.824i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (-14.6 + 30.5i)T + (-4.09e3 - 5.12e3i)T^{2} \) |
| 5 | \( 1 + (-1.12e3 - 257. i)T + (3.51e5 + 1.69e5i)T^{2} \) |
| 7 | \( 1 + 4.07e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-6.05e3 - 7.59e3i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (9.96e3 - 4.36e4i)T + (-7.34e8 - 3.53e8i)T^{2} \) |
| 17 | \( 1 + (1.22e4 + 5.37e4i)T + (-6.28e9 + 3.02e9i)T^{2} \) |
| 19 | \( 1 + (6.73e4 + 5.36e4i)T + (3.77e9 + 1.65e10i)T^{2} \) |
| 23 | \( 1 + (-8.62e4 - 1.08e5i)T + (-1.74e10 + 7.63e10i)T^{2} \) |
| 29 | \( 1 + (4.62e5 + 9.59e5i)T + (-3.11e11 + 3.91e11i)T^{2} \) |
| 31 | \( 1 + (-1.11e4 + 5.38e3i)T + (5.31e11 - 6.66e11i)T^{2} \) |
| 37 | \( 1 - 3.06e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (1.79e6 - 8.64e5i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-2.42e5 + 3.03e5i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (2.54e6 + 1.11e7i)T + (-5.60e13 + 2.70e13i)T^{2} \) |
| 59 | \( 1 + (-2.19e6 - 9.60e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (4.86e6 - 1.01e7i)T + (-1.19e14 - 1.49e14i)T^{2} \) |
| 67 | \( 1 + (-4.56e6 + 5.72e6i)T + (-9.03e13 - 3.95e14i)T^{2} \) |
| 71 | \( 1 + (-2.44e7 - 1.95e7i)T + (1.43e14 + 6.29e14i)T^{2} \) |
| 73 | \( 1 + (1.11e7 + 2.54e6i)T + (7.26e14 + 3.49e14i)T^{2} \) |
| 79 | \( 1 - 2.94e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.74e7 + 1.31e7i)T + (1.40e15 + 1.76e15i)T^{2} \) |
| 89 | \( 1 + (2.49e7 - 5.18e7i)T + (-2.45e15 - 3.07e15i)T^{2} \) |
| 97 | \( 1 + (-3.64e6 - 4.56e6i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−13.73656480929336121124922035455, −13.49940614827519788241536310970, −11.32703795816394225959722930475, −10.15450038375756846934523787403, −9.647118336945244834406550463108, −7.04298696738199511262780768669, −6.63531096339725991096492431756, −4.76722422420461530085517227711, −2.17169474631913782922689298216, −1.36807769510580235475316539075,
1.84311760230958314328571069077, 3.07234376893122520235004760516, 5.46462495495914395977323359738, 6.36972363914231104110812554160, 8.530950739735167683767796001757, 9.304621568760791870164146207800, 10.63138278227626361427061015038, 12.55396156132282930283678666759, 12.73894036703379095585764968756, 14.61709091703878889098020948866
