L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 4.24i·7-s + 2.82i·8-s − 1.41i·11-s + 4.24i·13-s + 6·14-s + 4.00·16-s − 2.82i·17-s − 4·19-s − 2.00·22-s − 6·23-s + 6·26-s − 8.48i·28-s − 6·29-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s + 1.60i·7-s + 1.00i·8-s − 0.426i·11-s + 1.17i·13-s + 1.60·14-s + 1.00·16-s − 0.685i·17-s − 0.917·19-s − 0.426·22-s − 1.25·23-s + 1.17·26-s − 1.60i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 1.41iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.943988129895287139199922873902, −8.454115931298661236787194428559, −7.36333165492618015195395587155, −6.02980566535409507150749377977, −5.58709075838795321300631670447, −4.47406466746149651242449255483, −3.67021397352078189010026446535, −2.38911572744647491431735255709, −1.97639686754233461993743636092, 0,
1.42092304265090400906119991156, 3.34560043828901261758983893304, 4.11181320787517250526126502702, 4.85393781871152502645944427568, 5.88354853880028814942318557493, 6.66949795278718274496167074821, 7.37236661811206806249157074321, 8.049317074485377000421313087423, 8.624068777247899082972725224282