| L(s) = 1 | + (−12.2 − 10.2i)2-s + (−246. + 89.8i)3-s + (44.4 + 252. i)4-s + (−0.584 + 3.31i)5-s + (3.94e3 + 1.43e3i)6-s + (1.73e3 + 3.01e3i)7-s + (2.04e3 − 3.54e3i)8-s + (3.77e4 − 3.16e4i)9-s + (41.2 − 34.6i)10-s + (−2.55e3 + 4.43e3i)11-s + (−3.36e4 − 5.82e4i)12-s + (1.52e5 + 5.55e4i)13-s + (9.66e3 − 5.47e4i)14-s + (−153. − 870. i)15-s + (−6.15e4 + 2.24e4i)16-s + (−2.43e5 − 2.03e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (−1.75 + 0.640i)3-s + (0.0868 + 0.492i)4-s + (−0.000418 + 0.00237i)5-s + (1.24 + 0.452i)6-s + (0.273 + 0.474i)7-s + (0.176 − 0.306i)8-s + (1.91 − 1.60i)9-s + (0.00130 − 0.00109i)10-s + (−0.0526 + 0.0912i)11-s + (−0.468 − 0.810i)12-s + (1.48 + 0.539i)13-s + (0.0672 − 0.381i)14-s + (−0.000783 − 0.00444i)15-s + (−0.234 + 0.0855i)16-s + (−0.705 − 0.592i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.112806 + 0.361126i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112806 + 0.361126i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (2.45e5 - 5.12e5i)T \) |
| good | 3 | \( 1 + (246. - 89.8i)T + (1.50e4 - 1.26e4i)T^{2} \) |
| 5 | \( 1 + (0.584 - 3.31i)T + (-1.83e6 - 6.68e5i)T^{2} \) |
| 7 | \( 1 + (-1.73e3 - 3.01e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.55e3 - 4.43e3i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.52e5 - 5.55e4i)T + (8.12e9 + 6.81e9i)T^{2} \) |
| 17 | \( 1 + (2.43e5 + 2.03e5i)T + (2.05e10 + 1.16e11i)T^{2} \) |
| 23 | \( 1 + (-2.09e5 - 1.18e6i)T + (-1.69e12 + 6.16e11i)T^{2} \) |
| 29 | \( 1 + (-2.72e5 + 2.28e5i)T + (2.51e12 - 1.42e13i)T^{2} \) |
| 31 | \( 1 + (2.25e6 + 3.90e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 - 3.86e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + (3.25e7 - 1.18e7i)T + (2.50e14 - 2.10e14i)T^{2} \) |
| 43 | \( 1 + (-2.11e6 + 1.20e7i)T + (-4.72e14 - 1.71e14i)T^{2} \) |
| 47 | \( 1 + (-7.84e6 + 6.58e6i)T + (1.94e14 - 1.10e15i)T^{2} \) |
| 53 | \( 1 + (7.13e6 + 4.04e7i)T + (-3.10e15 + 1.12e15i)T^{2} \) |
| 59 | \( 1 + (2.37e7 + 1.99e7i)T + (1.50e15 + 8.53e15i)T^{2} \) |
| 61 | \( 1 + (-3.30e7 - 1.87e8i)T + (-1.09e16 + 3.99e15i)T^{2} \) |
| 67 | \( 1 + (1.10e8 - 9.30e7i)T + (4.72e15 - 2.67e16i)T^{2} \) |
| 71 | \( 1 + (5.25e7 - 2.98e8i)T + (-4.30e16 - 1.56e16i)T^{2} \) |
| 73 | \( 1 + (-1.56e8 + 5.69e7i)T + (4.50e16 - 3.78e16i)T^{2} \) |
| 79 | \( 1 + (4.42e8 - 1.60e8i)T + (9.18e16 - 7.70e16i)T^{2} \) |
| 83 | \( 1 + (-1.15e8 - 2.00e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 + (7.75e8 + 2.82e8i)T + (2.68e17 + 2.25e17i)T^{2} \) |
| 97 | \( 1 + (3.97e8 + 3.33e8i)T + (1.32e17 + 7.48e17i)T^{2} \) |
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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
−15.28025347604984156940999365481, −13.13018637966488614460302907977, −11.77306028172411584943616021630, −11.23040975353144770612345013226, −10.16075757532803457645846375129, −8.844825833193578018171229996131, −6.74930135060401267757510848588, −5.47012088467460533326786064021, −3.98410181320833143045968113727, −1.34490893347808387745475963491,
0.24826414004176404765039459531, 1.36246959065171613519304502343, 4.74257531208408516546546525099, 6.12683681488110639637748412233, 6.94867594008709302383949616282, 8.472076583428330802383064336099, 10.71418406325200346358894901555, 10.91672215183748673452903349232, 12.52499563926262704126327668544, 13.57524662226488708077390481564
