| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 4·7-s + 3·8-s + 9-s − 4·11-s − 12-s + 2·13-s − 4·14-s − 16-s − 18-s + 4·19-s + 4·21-s + 4·22-s + 4·23-s + 3·24-s − 5·25-s − 2·26-s + 27-s − 4·28-s − 4·31-s − 5·32-s − 4·33-s − 36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 1.06·14-s − 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.872·21-s + 0.852·22-s + 0.834·23-s + 0.612·24-s − 25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s − 0.718·31-s − 0.883·32-s − 0.696·33-s − 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.350883486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350883486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−9.933534996928952467173977903436, −9.294392762170980253636920587383, −8.255761477500213584777769529872, −7.998273498275412381565667741916, −7.25491104135417288767198138812, −5.53430834097131793284892772151, −4.84950601616688663911919178631, −3.83125266908850693476257317433, −2.33338817193266243324550750634, −1.10822312656348596804753950237,
1.10822312656348596804753950237, 2.33338817193266243324550750634, 3.83125266908850693476257317433, 4.84950601616688663911919178631, 5.53430834097131793284892772151, 7.25491104135417288767198138812, 7.998273498275412381565667741916, 8.255761477500213584777769529872, 9.294392762170980253636920587383, 9.933534996928952467173977903436
