| L(s) = 1 | − 1.93i·5-s − 3.93i·7-s − 2.03·11-s − 2.46·13-s − 4.76i·17-s − 6.81i·19-s + 7.59·23-s + 1.26·25-s + 2.31i·29-s − 2.87i·31-s − 7.59·35-s − 3.73·37-s − 3.20i·41-s + 3.93i·43-s + 5.56·47-s + ⋯ |
| L(s) = 1 | − 0.863i·5-s − 1.48i·7-s − 0.613·11-s − 0.683·13-s − 1.15i·17-s − 1.56i·19-s + 1.58·23-s + 0.253·25-s + 0.429i·29-s − 0.517i·31-s − 1.28·35-s − 0.613·37-s − 0.500i·41-s + 0.599i·43-s + 0.811·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.374973250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.374973250\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.93iT - 5T^{2} \) |
| 7 | \( 1 + 3.93iT - 7T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 4.76iT - 17T^{2} \) |
| 19 | \( 1 + 6.81iT - 19T^{2} \) |
| 23 | \( 1 - 7.59T + 23T^{2} \) |
| 29 | \( 1 - 2.31iT - 29T^{2} \) |
| 31 | \( 1 + 2.87iT - 31T^{2} \) |
| 37 | \( 1 + 3.73T + 37T^{2} \) |
| 41 | \( 1 + 3.20iT - 41T^{2} \) |
| 43 | \( 1 - 3.93iT - 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 5.56T + 59T^{2} \) |
| 61 | \( 1 - 5.73T + 61T^{2} \) |
| 67 | \( 1 + 3.93iT - 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 1.05iT - 79T^{2} \) |
| 83 | \( 1 - 4.07T + 83T^{2} \) |
| 89 | \( 1 + 3.62iT - 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
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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
−7.68037654554267146493742951143, −7.12310269589788774573246170863, −6.77677960129296655648387388176, −5.29636008616171187339302360046, −4.93956381690161368430330188925, −4.35291640772200097084680006133, −3.26807872176677025339529006960, −2.45482195008399249063886657016, −1.02147561399620629105075894009, −0.41531188006269512539983873208,
1.57847406417961957438476853152, 2.55379006693510341173056722234, 3.03156989336735795545906120311, 4.04226270472832943968978421527, 5.16462423786034500927808048780, 5.66175573208089908125572725494, 6.35263741077295308955854519972, 7.13909129472088323972458171538, 7.85640986242025093421536339405, 8.621762614834595656452488934655
