L(s) = 1 | + (−0.160 − 1.40i)2-s − i·3-s + (−1.94 + 0.451i)4-s + i·5-s + (−1.40 + 0.160i)6-s + 1.44·7-s + (0.947 + 2.66i)8-s − 9-s + (1.40 − 0.160i)10-s − 4.31·11-s + (0.451 + 1.94i)12-s + 5.03·13-s + (−0.232 − 2.03i)14-s + 15-s + (3.59 − 1.75i)16-s − 3.32i·17-s + ⋯ |
L(s) = 1 | + (−0.113 − 0.993i)2-s − 0.577i·3-s + (−0.974 + 0.225i)4-s + 0.447i·5-s + (−0.573 + 0.0655i)6-s + 0.546·7-s + (0.334 + 0.942i)8-s − 0.333·9-s + (0.444 − 0.0507i)10-s − 1.30·11-s + (0.130 + 0.562i)12-s + 1.39·13-s + (−0.0620 − 0.543i)14-s + 0.258·15-s + (0.898 − 0.439i)16-s − 0.806i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6192152930\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6192152930\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.160 + 1.40i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (3.36 + 3.41i)T \) |
good | 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 + 4.31T + 11T^{2} \) |
| 13 | \( 1 - 5.03T + 13T^{2} \) |
| 17 | \( 1 + 3.32iT - 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 29 | \( 1 - 0.0416T + 29T^{2} \) |
| 31 | \( 1 + 4.91iT - 31T^{2} \) |
| 37 | \( 1 + 7.69iT - 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 1.31T + 43T^{2} \) |
| 47 | \( 1 - 3.27iT - 47T^{2} \) |
| 53 | \( 1 - 3.82iT - 53T^{2} \) |
| 59 | \( 1 + 9.30iT - 59T^{2} \) |
| 61 | \( 1 + 3.55iT - 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 + 2.13iT - 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 + 8.18T + 83T^{2} \) |
| 89 | \( 1 - 9.95iT - 89T^{2} \) |
| 97 | \( 1 - 0.397iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.097789861771261043238227006212, −8.192930313889544502859188488618, −7.914311278135791173459884315919, −6.63526595754802275969945004018, −5.70021798863052525871080248959, −4.71927697535968945920865916678, −3.69565069707243756759633207288, −2.59494323121906509712236694573, −1.83307730829071051815954919311, −0.25946725054707871413376511442,
1.61647396161162041783798739345, 3.45039518560830319654518409087, 4.35432490171784222510810792593, 5.13057687299829639158308226913, 5.86988984776112613689605665710, 6.67337508506601338234642822308, 7.976372932422508411107016548244, 8.368317606148891307112430656880, 8.871251223432729752932915979336, 10.14838918992560907849537119339