L(s) = 1 | + (−1.30 − 0.541i)2-s + (0.541 − 1.64i)3-s + (1.41 + 1.41i)4-s + (1.25 + 1.84i)5-s + (−1.59 + 1.85i)6-s + 3.29·7-s + (−1.08 − 2.61i)8-s + (−2.41 − 1.78i)9-s + (−0.645 − 3.09i)10-s − 2.51i·11-s + (3.09 − 1.56i)12-s − 4.65·13-s + (−4.29 − 1.78i)14-s + (3.72 − 1.07i)15-s + 4i·16-s + 3.69·17-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (0.312 − 0.949i)3-s + (0.707 + 0.707i)4-s + (0.563 + 0.826i)5-s + (−0.652 + 0.758i)6-s + 1.24·7-s + (−0.382 − 0.923i)8-s + (−0.804 − 0.593i)9-s + (−0.204 − 0.978i)10-s − 0.759i·11-s + (0.892 − 0.450i)12-s − 1.29·13-s + (−1.14 − 0.475i)14-s + (0.960 − 0.276i)15-s + i·16-s + 0.896·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.803040 - 0.386757i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.803040 - 0.386757i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.30 + 0.541i)T \) |
| 3 | \( 1 + (-0.541 + 1.64i)T \) |
| 5 | \( 1 + (-1.25 - 1.84i)T \) |
good | 7 | \( 1 - 3.29T + 7T^{2} \) |
| 11 | \( 1 + 2.51iT - 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 2.61iT - 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 1.17iT - 31T^{2} \) |
| 37 | \( 1 + 1.92T + 37T^{2} \) |
| 41 | \( 1 - 8.59iT - 41T^{2} \) |
| 43 | \( 1 + 6.01iT - 43T^{2} \) |
| 47 | \( 1 + 2.61iT - 47T^{2} \) |
| 53 | \( 1 - 4.59iT - 53T^{2} \) |
| 59 | \( 1 - 2.51iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 3.29iT - 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 6.58iT - 73T^{2} \) |
| 79 | \( 1 + 16.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.37T + 83T^{2} \) |
| 89 | \( 1 + 5.03iT - 89T^{2} \) |
| 97 | \( 1 - 2.72iT - 97T^{2} \) |
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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
−13.29948416996291836348006270976, −11.97870387752774676354175967517, −11.33799955974176083622512128209, −10.19788936866637195945396117114, −9.018234130715638059539196703076, −7.80881104697005118075569315287, −7.22162352363267510317866073168, −5.74269853069801206264339645395, −3.06203263807475550417502343270, −1.73610263099803189247974954658,
2.07608496722704828197574997567, 4.73789409013169917097316458723, 5.49345642072499723526925121580, 7.50829940496155040846872623720, 8.402629662360313603637777139628, 9.480900968872376155021201484544, 10.07332611226343479010296373145, 11.26131003303173084221039094904, 12.38574940154574820563031536047, 14.21766596236399480250282812567