| L(s) = 1 | + (−9.79 − 5.65i)2-s + (65.0 + 37.5i)3-s + (63.9 + 110. i)4-s + (393. − 681. i)5-s + (−424. − 735. i)6-s − 1.84e3·7-s − 1.44e3i·8-s + (−463. − 802. i)9-s + (−7.71e3 + 4.45e3i)10-s + 1.28e4·11-s + 9.60e3i·12-s + (−2.95e3 + 1.70e3i)13-s + (1.81e4 + 1.04e4i)14-s + (5.11e4 − 2.95e4i)15-s + (−8.19e3 + 1.41e4i)16-s + (3.00e4 − 5.20e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.802 + 0.463i)3-s + (0.249 + 0.433i)4-s + (0.629 − 1.09i)5-s + (−0.327 − 0.567i)6-s − 0.769·7-s − 0.353i·8-s + (−0.0706 − 0.122i)9-s + (−0.771 + 0.445i)10-s + 0.876·11-s + 0.463i·12-s + (−0.103 + 0.0596i)13-s + (0.471 + 0.272i)14-s + (1.01 − 0.583i)15-s + (−0.125 + 0.216i)16-s + (0.360 − 0.623i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0303 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0303 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.13710 - 1.17215i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13710 - 1.17215i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (9.79 + 5.65i)T \) |
| 19 | \( 1 + (1.03e5 + 7.90e4i)T \) |
| good | 3 | \( 1 + (-65.0 - 37.5i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-393. + 681. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + 1.84e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.28e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (2.95e3 - 1.70e3i)T + (4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + (-3.00e4 + 5.20e4i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 23 | \( 1 + (1.40e5 + 2.43e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-9.23e5 + 5.33e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + 1.27e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.07e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-1.35e6 - 7.81e5i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.53e6 + 2.66e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-5.48e5 - 9.50e5i)T + (-1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (5.01e6 - 2.89e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.37e6 - 7.96e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-6.87e6 - 1.19e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-7.43e6 + 4.29e6i)T + (2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + (-1.74e7 - 1.00e7i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.18e7 - 2.04e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-7.96e6 - 4.60e6i)T + (7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 7.51e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (5.37e7 - 3.10e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.04e8 - 6.01e7i)T + (3.91e15 + 6.78e15i)T^{2} \) |
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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
−14.16181406931103919906890292605, −12.98496316761107883286011738703, −11.85529685526084056766312661275, −9.940606966268567196058377403164, −9.281076564356379122257082279576, −8.401636272756223347376166598572, −6.36154723567807357510356234688, −4.24831207807687098616301854921, −2.58479867040157480394751687972, −0.72465186099065226578462317410,
1.78351442747447187232458136980, 3.20066591869235548239641311986, 6.10277524165745430859246783988, 7.07598462010725319126522734852, 8.468983752444338945804030334225, 9.742182583683760958458672784523, 10.78375679331926150383961914128, 12.61979910099230359079057843597, 14.15065658604920302570614838208, 14.48480756305142700785292078220
