| L(s) = 1 | − 5-s − 3·7-s + 5·11-s + 5·13-s + 2·17-s + 4·19-s + 23-s − 4·25-s − 9·29-s − 31-s + 3·35-s + 6·37-s − 3·41-s − 43-s + 3·47-s + 2·49-s + 2·53-s − 5·55-s + 11·59-s − 7·61-s − 5·65-s + 67-s − 4·71-s − 2·73-s − 15·77-s + 79-s + 83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.50·11-s + 1.38·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s − 4/5·25-s − 1.67·29-s − 0.179·31-s + 0.507·35-s + 0.986·37-s − 0.468·41-s − 0.152·43-s + 0.437·47-s + 2/7·49-s + 0.274·53-s − 0.674·55-s + 1.43·59-s − 0.896·61-s − 0.620·65-s + 0.122·67-s − 0.474·71-s − 0.234·73-s − 1.70·77-s + 0.112·79-s + 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.777963294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.777963294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| 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
−8.206076036624544607586809127025, −7.42550541051641200575881013466, −6.75429776792018770227605635281, −6.06543866952929738450715955924, −5.55310591278784440490443142611, −4.21105162226506464563979927553, −3.65505252719295979022106964600, −3.19738780610208031359073353034, −1.74327218236144124349350390954, −0.75209062743662468140547933595,
0.75209062743662468140547933595, 1.74327218236144124349350390954, 3.19738780610208031359073353034, 3.65505252719295979022106964600, 4.21105162226506464563979927553, 5.55310591278784440490443142611, 6.06543866952929738450715955924, 6.75429776792018770227605635281, 7.42550541051641200575881013466, 8.206076036624544607586809127025
