| L(s) = 1 | + 4i·2-s + 2i·3-s − 8·4-s − 8·6-s − 6i·7-s + 23·9-s + 32·11-s − 16i·12-s − 38i·13-s + 24·14-s − 64·16-s − 26i·17-s + 92i·18-s − 100·19-s + 12·21-s + 128i·22-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s + 0.384i·3-s − 4-s − 0.544·6-s − 0.323i·7-s + 0.851·9-s + 0.877·11-s − 0.384i·12-s − 0.810i·13-s + 0.458·14-s − 16-s − 0.370i·17-s + 1.20i·18-s − 1.20·19-s + 0.124·21-s + 1.24i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.619805 + 1.00286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.619805 + 1.00286i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 4iT - 8T^{2} \) |
| 3 | \( 1 - 2iT - 27T^{2} \) |
| 7 | \( 1 + 6iT - 343T^{2} \) |
| 11 | \( 1 - 32T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 26iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 50T + 2.43e4T^{2} \) |
| 31 | \( 1 + 108T + 2.97e4T^{2} \) |
| 37 | \( 1 + 266iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 22T + 6.89e4T^{2} \) |
| 43 | \( 1 - 442iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 514iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 2iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 500T + 2.05e5T^{2} \) |
| 61 | \( 1 + 518T + 2.26e5T^{2} \) |
| 67 | \( 1 + 126iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 412T + 3.57e5T^{2} \) |
| 73 | \( 1 + 878iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 600T + 4.93e5T^{2} \) |
| 83 | \( 1 - 282iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 150T + 7.04e5T^{2} \) |
| 97 | \( 1 + 386iT - 9.12e5T^{2} \) |
| 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
−17.11743215577945325806715269658, −16.20934082629795375547420259254, −15.19206150308486621121931816072, −14.25848234097608282832566463691, −12.75134354867631497106770318456, −10.76363079090156917444942535885, −9.146024258147395818482938127457, −7.61870573183649997930724515184, −6.30450370902657317195869445350, −4.48542669227441876780418770748,
1.77473224837234084161377035457, 4.05739679486664143625222420840, 6.76866763762843534060552480768, 8.978158478624694787817787261191, 10.28520797796661043214428666792, 11.66210908476758601098263422867, 12.54274079742628837608247720575, 13.69634233116587949986292753343, 15.35278195704156773370048623723, 16.98833288212420698173490086400
