L(s) = 1 | − 1.41i·5-s − 2.82i·7-s + 4·11-s + 2·13-s + 1.41i·17-s + 5.65i·19-s + 4·23-s + 2.99·25-s − 7.07i·29-s + 8.48i·31-s − 4.00·35-s + 8·37-s + 4.24i·41-s − 11.3i·43-s − 12·47-s + ⋯ |
L(s) = 1 | − 0.632i·5-s − 1.06i·7-s + 1.20·11-s + 0.554·13-s + 0.342i·17-s + 1.29i·19-s + 0.834·23-s + 0.599·25-s − 1.31i·29-s + 1.52i·31-s − 0.676·35-s + 1.31·37-s + 0.662i·41-s − 1.72i·43-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.038506609\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.038506609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 15.5iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.671132159022916897196033894620, −8.352174433618556756146477667705, −7.26508398200434080060145066892, −6.62178011092317198411646866205, −5.81178760462294483788582839207, −4.74871547394549170034787114716, −4.01354161046819685313686401912, −3.34591889857637317195337084234, −1.66296134316450211293625620685, −0.876164686237853542763896459170,
1.16084756680104319818946279339, 2.52494194985834679533180471482, 3.16424180665631368706856515975, 4.31004380098127609361863092683, 5.18977828004671580578011363272, 6.19019991054668863312000503695, 6.65999770087998053024576615956, 7.49112417903938353981118073297, 8.539105003596820625964827664326, 9.182796358858131062005746422852