L(s) = 1 | − 2i·2-s − 1.87·3-s − 4·4-s − 8.73i·5-s + 3.74i·6-s − 6.27·7-s + 8i·8-s − 23.5·9-s − 17.4·10-s − 45.9·11-s + 7.48·12-s − 7.07i·13-s + 12.5i·14-s + 16.3i·15-s + 16·16-s − 36.5i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.359·3-s − 0.5·4-s − 0.781i·5-s + 0.254i·6-s − 0.338·7-s + 0.353i·8-s − 0.870·9-s − 0.552·10-s − 1.25·11-s + 0.179·12-s − 0.150i·13-s + 0.239i·14-s + 0.281i·15-s + 0.250·16-s − 0.521i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0341148 + 0.547453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0341148 + 0.547453i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 37 | \( 1 + (-27.9 + 223. i)T \) |
good | 3 | \( 1 + 1.87T + 27T^{2} \) |
| 5 | \( 1 + 8.73iT - 125T^{2} \) |
| 7 | \( 1 + 6.27T + 343T^{2} \) |
| 11 | \( 1 + 45.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.07iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 36.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 20.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 89.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 223. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 18.0iT - 2.97e4T^{2} \) |
| 41 | \( 1 - 101.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 255. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 288.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 211. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 178. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 773.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 288.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 299. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 490.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 306. iT - 9.12e5T^{2} \) |
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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
−13.17799894446404761758795756509, −12.43796672551720531753430674760, −11.32912011734373793134248771617, −10.28367526247995025261508414942, −9.039923512835522355221129350037, −7.924571441381182997584641674647, −5.87390003091702978284351591934, −4.71240654897176763884341170451, −2.74557956493813177053342022186, −0.35307995440977658816849724499,
3.08140333921697894308466505557, 5.16923778750482219107148364834, 6.30741301579360555753406099915, 7.49477314386140055239284830784, 8.772036873729357876042294757526, 10.26114284922037717162578904541, 11.17509939690156041498997231998, 12.62527401542017348057375198579, 13.73784179163372344456435203110, 14.73722198828532263313578090451