| L(s) = 1 | + 8i·2-s − 64·4-s + 1.57e3i·7-s − 512i·8-s − 7.33e3·11-s − 3.80e3i·13-s − 1.26e4·14-s + 4.09e3·16-s − 6.60e3i·17-s − 2.48e4·19-s − 5.86e4i·22-s − 4.14e4i·23-s + 3.04e4·26-s − 1.00e5i·28-s − 4.16e4·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.73i·7-s − 0.353i·8-s − 1.66·11-s − 0.479i·13-s − 1.22·14-s + 0.250·16-s − 0.326i·17-s − 0.831·19-s − 1.17i·22-s − 0.710i·23-s + 0.339·26-s − 0.868i·28-s − 0.316·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9943519734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9943519734\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.57e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 7.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.80e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 6.60e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.48e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.14e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 4.16e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.31e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.64e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.39e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.56e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 4.33e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 7.86e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 7.45e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.66e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.29e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.71e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.65e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 3.80e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.22e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 5.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.06e6iT - 8.07e13T^{2} \) |
| 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
−9.870551785775113183297661690816, −8.781151159053438989069387018886, −8.294354459482063565940885453536, −7.32325276591825146328555814832, −6.03610977874601646493690715834, −5.49839058619803837123991131402, −4.61172325675250917613826892015, −2.94316259186136104533990340652, −2.20597150265508362997617336115, −0.31180785411366247782117773141,
0.60459855960892655313282554907, 1.75351246138996750571353724991, 2.96876795608612826832244639083, 4.05782610358895608235741221583, 4.77652025277288387564299427854, 6.07841715827662923953595956925, 7.40485057451917420765872999395, 7.899112929550388582366076348641, 9.158864629807141752400276745086, 10.22208357599483195796992329383
