| L(s) = 1 | − 3-s − 2·7-s + 9-s + 6·11-s + 2·13-s + 2·17-s + 2·19-s + 2·21-s + 6·23-s − 27-s + 10·29-s + 4·31-s − 6·33-s − 2·37-s − 2·39-s + 6·41-s + 6·47-s − 3·49-s − 2·51-s + 10·53-s − 2·57-s − 6·59-s + 6·61-s − 2·63-s − 16·67-s − 6·69-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s + 1.85·29-s + 0.718·31-s − 1.04·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 1.37·53-s − 0.264·57-s − 0.781·59-s + 0.768·61-s − 0.251·63-s − 1.95·67-s − 0.722·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153212635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153212635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.36578946830678992873502736325, −6.98845411105128296815967194479, −6.13982127122699995474080210075, −6.00138165911262773076839054929, −4.84741679678121519199482059875, −4.24789754651108531948912759266, −3.44427436028444049979425298139, −2.78744640089893360011919083708, −1.35163666357043567693657538985, −0.843225383026201512029540897959,
0.843225383026201512029540897959, 1.35163666357043567693657538985, 2.78744640089893360011919083708, 3.44427436028444049979425298139, 4.24789754651108531948912759266, 4.84741679678121519199482059875, 6.00138165911262773076839054929, 6.13982127122699995474080210075, 6.98845411105128296815967194479, 7.36578946830678992873502736325
