| L(s) = 1 | + 3-s − 1.87·5-s − 0.584·7-s + 9-s + 5.86·11-s − 13-s − 1.87·15-s − 5.53·17-s + 4.96·19-s − 0.584·21-s − 8.72·23-s − 1.47·25-s + 27-s − 5.17·29-s + 7.04·31-s + 5.86·33-s + 1.09·35-s + 0.398·37-s − 39-s + 5.30·41-s + 7.13·43-s − 1.87·45-s − 0.461·47-s − 6.65·49-s − 5.53·51-s + 0.0988·53-s − 11.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.839·5-s − 0.220·7-s + 0.333·9-s + 1.76·11-s − 0.277·13-s − 0.484·15-s − 1.34·17-s + 1.13·19-s − 0.127·21-s − 1.82·23-s − 0.295·25-s + 0.192·27-s − 0.961·29-s + 1.26·31-s + 1.02·33-s + 0.185·35-s + 0.0655·37-s − 0.160·39-s + 0.828·41-s + 1.08·43-s − 0.279·45-s − 0.0672·47-s − 0.951·49-s − 0.774·51-s + 0.0135·53-s − 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.078508354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078508354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 + 0.584T + 7T^{2} \) |
| 11 | \( 1 - 5.86T + 11T^{2} \) |
| 17 | \( 1 + 5.53T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 23 | \( 1 + 8.72T + 23T^{2} \) |
| 29 | \( 1 + 5.17T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 - 0.398T + 37T^{2} \) |
| 41 | \( 1 - 5.30T + 41T^{2} \) |
| 43 | \( 1 - 7.13T + 43T^{2} \) |
| 47 | \( 1 + 0.461T + 47T^{2} \) |
| 53 | \( 1 - 0.0988T + 53T^{2} \) |
| 59 | \( 1 - 9.74T + 59T^{2} \) |
| 61 | \( 1 - 3.50T + 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 5.96T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
| 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
−7.79025393897285660065570341675, −6.95780251921863011775339115324, −6.49505042093738592915231934874, −5.71803103226287633307045519737, −4.61565254477551639611293154085, −3.90873222790938776414551587200, −3.75011486161004477029071654509, −2.59204349904139893037721447368, −1.78919655310865994510880122448, −0.66383376683142703338850732955,
0.66383376683142703338850732955, 1.78919655310865994510880122448, 2.59204349904139893037721447368, 3.75011486161004477029071654509, 3.90873222790938776414551587200, 4.61565254477551639611293154085, 5.71803103226287633307045519737, 6.49505042093738592915231934874, 6.95780251921863011775339115324, 7.79025393897285660065570341675
