L(s) = 1 | − 0.445·2-s − 0.801·4-s + 7-s + 0.801·8-s + 9-s − 1.80·11-s − 0.445·14-s + 0.445·16-s − 0.445·18-s + 0.801·22-s + 1.24·23-s + 25-s − 0.801·28-s − 0.445·29-s − 32-s − 0.801·36-s + 1.24·37-s − 0.445·43-s + 1.44·44-s − 0.554·46-s + 49-s − 0.445·50-s + 1.24·53-s + 0.801·56-s + 0.198·58-s + 63-s + 1.24·67-s + ⋯ |
L(s) = 1 | − 0.445·2-s − 0.801·4-s + 7-s + 0.801·8-s + 9-s − 1.80·11-s − 0.445·14-s + 0.445·16-s − 0.445·18-s + 0.801·22-s + 1.24·23-s + 25-s − 0.801·28-s − 0.445·29-s − 32-s − 0.801·36-s + 1.24·37-s − 0.445·43-s + 1.44·44-s − 0.554·46-s + 49-s − 0.445·50-s + 1.24·53-s + 0.801·56-s + 0.198·58-s + 63-s + 1.24·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7916881147\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7916881147\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.445T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 - 1.24T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.991745425586182789695086044787, −9.119657318418242066167021450677, −8.266596467939380563377268633280, −7.72130926096950090002480640264, −6.97643274595197493095497685454, −5.34724573914641545646938486445, −4.95463230479145256427190738211, −4.04564808194053864952861860423, −2.57310065318110210323442610795, −1.16575599327724355573868171476,
1.16575599327724355573868171476, 2.57310065318110210323442610795, 4.04564808194053864952861860423, 4.95463230479145256427190738211, 5.34724573914641545646938486445, 6.97643274595197493095497685454, 7.72130926096950090002480640264, 8.266596467939380563377268633280, 9.119657318418242066167021450677, 9.991745425586182789695086044787