| L(s) = 1 | + (12.2 − 10.2i)2-s + (−29.9 − 10.8i)3-s + (44.4 − 252. i)4-s + (−197. − 1.12e3i)5-s + (−478. + 174. i)6-s + (−3.74e3 + 6.48e3i)7-s + (−2.04e3 − 3.54e3i)8-s + (−1.43e4 − 1.20e4i)9-s + (−1.39e4 − 1.17e4i)10-s + (2.39e4 + 4.14e4i)11-s + (−4.07e3 + 7.05e3i)12-s + (1.77e4 − 6.45e3i)13-s + (2.08e4 + 1.18e5i)14-s + (−6.29e3 + 3.56e4i)15-s + (−6.15e4 − 2.24e4i)16-s + (−1.74e5 + 1.46e5i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.213 − 0.0776i)3-s + (0.0868 − 0.492i)4-s + (−0.141 − 0.801i)5-s + (−0.150 + 0.0548i)6-s + (−0.589 + 1.02i)7-s + (−0.176 − 0.306i)8-s + (−0.726 − 0.609i)9-s + (−0.441 − 0.370i)10-s + (0.492 + 0.852i)11-s + (−0.0567 + 0.0982i)12-s + (0.172 − 0.0626i)13-s + (0.144 + 0.821i)14-s + (−0.0320 + 0.181i)15-s + (−0.234 − 0.0855i)16-s + (−0.505 + 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.170056 + 0.259093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170056 + 0.259093i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-12.2 + 10.2i)T \) |
| 19 | \( 1 + (5.50e5 - 1.40e5i)T \) |
| good | 3 | \( 1 + (29.9 + 10.8i)T + (1.50e4 + 1.26e4i)T^{2} \) |
| 5 | \( 1 + (197. + 1.12e3i)T + (-1.83e6 + 6.68e5i)T^{2} \) |
| 7 | \( 1 + (3.74e3 - 6.48e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.39e4 - 4.14e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.77e4 + 6.45e3i)T + (8.12e9 - 6.81e9i)T^{2} \) |
| 17 | \( 1 + (1.74e5 - 1.46e5i)T + (2.05e10 - 1.16e11i)T^{2} \) |
| 23 | \( 1 + (4.36e5 - 2.47e6i)T + (-1.69e12 - 6.16e11i)T^{2} \) |
| 29 | \( 1 + (3.04e6 + 2.55e6i)T + (2.51e12 + 1.42e13i)T^{2} \) |
| 31 | \( 1 + (2.75e6 - 4.77e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 5.80e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.15e7 + 7.86e6i)T + (2.50e14 + 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-1.48e6 - 8.39e6i)T + (-4.72e14 + 1.71e14i)T^{2} \) |
| 47 | \( 1 + (3.66e7 + 3.07e7i)T + (1.94e14 + 1.10e15i)T^{2} \) |
| 53 | \( 1 + (-1.55e7 + 8.84e7i)T + (-3.10e15 - 1.12e15i)T^{2} \) |
| 59 | \( 1 + (-4.60e7 + 3.86e7i)T + (1.50e15 - 8.53e15i)T^{2} \) |
| 61 | \( 1 + (-1.94e7 + 1.10e8i)T + (-1.09e16 - 3.99e15i)T^{2} \) |
| 67 | \( 1 + (2.10e8 + 1.76e8i)T + (4.72e15 + 2.67e16i)T^{2} \) |
| 71 | \( 1 + (-1.52e7 - 8.64e7i)T + (-4.30e16 + 1.56e16i)T^{2} \) |
| 73 | \( 1 + (-2.06e8 - 7.51e7i)T + (4.50e16 + 3.78e16i)T^{2} \) |
| 79 | \( 1 + (3.16e8 + 1.15e8i)T + (9.18e16 + 7.70e16i)T^{2} \) |
| 83 | \( 1 + (1.72e7 - 2.99e7i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (8.61e8 - 3.13e8i)T + (2.68e17 - 2.25e17i)T^{2} \) |
| 97 | \( 1 + (-9.91e7 + 8.31e7i)T + (1.32e17 - 7.48e17i)T^{2} \) |
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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
−14.74374051345029648226066316466, −13.09488016629907473755506101976, −12.31656265042718133322068229420, −11.45819656825522911944586458392, −9.648362667958889830111173006513, −8.657943562007545333137493106096, −6.45067673948211791045048540199, −5.27637857518591419117335526682, −3.63741139271264425299683384160, −1.83401180521163607754520993781,
0.091032982653525707556139173951, 2.86347478295656945539219546875, 4.26395585985026598740326984044, 6.10706858460399680019421050348, 7.06937030356251418072456935602, 8.614695727198214508899263100115, 10.56831278881152784608364526211, 11.33203649411345902907376325689, 13.04573824496719584842122718586, 14.01186720586967507901864104019
