| L(s) = 1 | − 3-s + 2.21·5-s + 2.90·7-s + 9-s + 11-s − 13-s − 2.21·15-s − 5.73·17-s + 4.14·19-s − 2.90·21-s − 5.33·23-s − 0.0967·25-s − 27-s − 6.49·29-s + 1.16·31-s − 33-s + 6.42·35-s + 10.2·37-s + 39-s − 2.14·41-s + 9.11·43-s + 2.21·45-s + 10.8·47-s + 1.42·49-s + 5.73·51-s − 6.42·53-s + 2.21·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.990·5-s + 1.09·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s − 0.571·15-s − 1.39·17-s + 0.951·19-s − 0.633·21-s − 1.11·23-s − 0.0193·25-s − 0.192·27-s − 1.20·29-s + 0.208·31-s − 0.174·33-s + 1.08·35-s + 1.68·37-s + 0.160·39-s − 0.335·41-s + 1.39·43-s + 0.330·45-s + 1.58·47-s + 0.204·49-s + 0.803·51-s − 0.883·53-s + 0.298·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.273977232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.273977232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.21T + 5T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 + 6.49T + 29T^{2} \) |
| 31 | \( 1 - 1.16T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 2.14T + 41T^{2} \) |
| 43 | \( 1 - 9.11T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 8.85T + 61T^{2} \) |
| 67 | \( 1 - 6.96T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 2.28T + 73T^{2} \) |
| 79 | \( 1 - 1.93T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 2.70T + 89T^{2} \) |
| 97 | \( 1 - 5.37T + 97T^{2} \) |
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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
−7.79712147084437839995946722624, −7.34961866663147795845981160940, −6.31039334479675584868382336780, −5.93008122337744306731775398962, −5.15798160540181297694261053968, −4.53808940547522898987032880377, −3.78016893848422068218780239864, −2.33042837691030566929083493794, −1.92074447105254444748780229061, −0.807427320376796933274759595252,
0.807427320376796933274759595252, 1.92074447105254444748780229061, 2.33042837691030566929083493794, 3.78016893848422068218780239864, 4.53808940547522898987032880377, 5.15798160540181297694261053968, 5.93008122337744306731775398962, 6.31039334479675584868382336780, 7.34961866663147795845981160940, 7.79712147084437839995946722624
