L(s) = 1 | − 3-s + 2·5-s + 7-s + 9-s + 5·11-s + 4·13-s − 2·15-s − 2·17-s − 21-s − 25-s − 27-s − 29-s + 3·31-s − 5·33-s + 2·35-s + 4·37-s − 4·39-s − 2·43-s + 2·45-s + 8·47-s − 6·49-s + 2·51-s + 9·53-s + 10·55-s + 7·59-s + 4·61-s + 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s − 0.516·15-s − 0.485·17-s − 0.218·21-s − 1/5·25-s − 0.192·27-s − 0.185·29-s + 0.538·31-s − 0.870·33-s + 0.338·35-s + 0.657·37-s − 0.640·39-s − 0.304·43-s + 0.298·45-s + 1.16·47-s − 6/7·49-s + 0.280·51-s + 1.23·53-s + 1.34·55-s + 0.911·59-s + 0.512·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.833109729\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.833109729\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−16.31034807325036, −15.92995392768628, −15.02182116515789, −14.66147283549408, −13.95391028670079, −13.39419936958970, −13.13435870457426, −12.04461182149160, −11.82190776666198, −11.13497366313442, −10.63871466078352, −9.928085466447988, −9.360773641906249, −8.804231395116648, −8.233377211300098, −7.240398434963280, −6.679479796421248, −5.982612574921075, −5.791461877646404, −4.748658304972288, −4.147464993010050, −3.463004611790009, −2.283404228804247, −1.548268233705654, −0.8565383517966445,
0.8565383517966445, 1.548268233705654, 2.283404228804247, 3.463004611790009, 4.147464993010050, 4.748658304972288, 5.791461877646404, 5.982612574921075, 6.679479796421248, 7.240398434963280, 8.233377211300098, 8.804231395116648, 9.360773641906249, 9.928085466447988, 10.63871466078352, 11.13497366313442, 11.82190776666198, 12.04461182149160, 13.13435870457426, 13.39419936958970, 13.95391028670079, 14.66147283549408, 15.02182116515789, 15.92995392768628, 16.31034807325036