L(s) = 1 | + (−0.819 − 0.819i)2-s + (1.37 − 1.37i)3-s − 0.655i·4-s + (2.23 − 0.0985i)5-s − 2.26·6-s + (−1.78 + 1.78i)7-s + (−2.17 + 2.17i)8-s − 0.799i·9-s + (−1.91 − 1.75i)10-s − 1.86i·11-s + (−0.903 − 0.903i)12-s + (1.52 − 1.52i)13-s + 2.92·14-s + (2.94 − 3.21i)15-s + 2.26·16-s + (−5.17 + 5.17i)17-s + ⋯ |
L(s) = 1 | + (−0.579 − 0.579i)2-s + (0.795 − 0.795i)3-s − 0.327i·4-s + (0.999 − 0.0440i)5-s − 0.922·6-s + (−0.674 + 0.674i)7-s + (−0.769 + 0.769i)8-s − 0.266i·9-s + (−0.604 − 0.553i)10-s − 0.561i·11-s + (−0.260 − 0.260i)12-s + (0.422 − 0.422i)13-s + 0.782·14-s + (0.759 − 0.830i)15-s + 0.565·16-s + (−1.25 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0271 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0271 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.745784 - 0.725827i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.745784 - 0.725827i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.23 + 0.0985i)T \) |
| 23 | \( 1 + (4.03 + 2.59i)T \) |
good | 2 | \( 1 + (0.819 + 0.819i)T + 2iT^{2} \) |
| 3 | \( 1 + (-1.37 + 1.37i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.78 - 1.78i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.86iT - 11T^{2} \) |
| 13 | \( 1 + (-1.52 + 1.52i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.17 - 5.17i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.69T + 19T^{2} \) |
| 29 | \( 1 - 4.99iT - 29T^{2} \) |
| 31 | \( 1 - 0.833T + 31T^{2} \) |
| 37 | \( 1 + (-3.00 + 3.00i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.310T + 41T^{2} \) |
| 43 | \( 1 + (2.17 + 2.17i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.19 + 8.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.21 - 6.21i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.19iT - 59T^{2} \) |
| 61 | \( 1 + 12.5iT - 61T^{2} \) |
| 67 | \( 1 + (1.14 - 1.14i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + (6.62 - 6.62i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.36T + 79T^{2} \) |
| 83 | \( 1 + (-0.256 - 0.256i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + (9.99 - 9.99i)T - 97iT^{2} \) |
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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
−13.29658376527488908609833642635, −12.47841301794313689604610180936, −11.00325017139577505554938611443, −10.06455805735942292155783930052, −8.958556011320459139867465188478, −8.403063343257777574922026900090, −6.53340251142697153095293486184, −5.62395504933120497185041745785, −2.87289013732522231965374798422, −1.75721139074267897435928259539,
2.93177684409792271680397832872, 4.31072284254326982915960208458, 6.31865582302191879598041748915, 7.29071020178361758594635137643, 8.711580508768401722633909605527, 9.607469472192368174330048112839, 9.928322692535220904802374754403, 11.69262766222869179045056460985, 13.23867000439848670618723961967, 13.78130391208601124761268289215