| L(s) = 1 | + (4 + 6.92i)2-s + (−31.9 + 55.4i)4-s + (−82.5 + 142. i)5-s + (254 + 439. i)7-s − 511.·8-s − 1.32e3·10-s + (1.51e3 + 2.61e3i)11-s + (−2.51e3 + 4.36e3i)13-s + (−2.03e3 + 3.51e3i)14-s + (−2.04e3 − 3.54e3i)16-s + 3.18e3·17-s + 1.50e3·19-s + (−5.28e3 − 9.14e3i)20-s + (−1.20e4 + 2.09e4i)22-s + (−3.78e4 + 6.54e4i)23-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.295 + 0.511i)5-s + (0.279 + 0.484i)7-s − 0.353·8-s − 0.417·10-s + (0.342 + 0.593i)11-s + (−0.318 + 0.550i)13-s + (−0.197 + 0.342i)14-s + (−0.125 − 0.216i)16-s + 0.157·17-s + 0.0504·19-s + (−0.147 − 0.255i)20-s + (−0.242 + 0.419i)22-s + (−0.647 + 1.12i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.093549655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093549655\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (82.5 - 142. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-254 - 439. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.51e3 - 2.61e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (2.51e3 - 4.36e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 3.18e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.50e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (3.78e4 - 6.54e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.13e4 + 7.15e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-8.74e4 + 1.51e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.54e5 - 2.66e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.68e5 + 2.91e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (1.91e5 + 3.31e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 7.60e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.11e6 - 1.92e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.12e6 + 1.94e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.36e5 - 1.27e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 5.00e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.51e6 + 6.08e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.32e6 + 2.29e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 6.77e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (8.08e6 + 1.40e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.03719472501806546594764978369, −11.49773930708098193623026249306, −10.02156404243106517775191882813, −9.007174433624170436474232443470, −7.78593507555900304019491927372, −6.96327138131373984480491156613, −5.80945513924447681515046601577, −4.61741762966938221303184905023, −3.41020324991272017036920459122, −1.89341792425272311339430149905,
0.25687553343924753765171036965, 1.37365187520319000128052991652, 2.98527733613566380975822936027, 4.20613829575332397818940125706, 5.18812855186794106330884525122, 6.54307534890021345054734628436, 7.991495633654315643108948712125, 8.892118997898237631337935538337, 10.19578830033338212369629005631, 10.93820172749231544170830503072
