| L(s) = 1 | − 3-s − 3·7-s + 9-s − 11-s + 13-s + 17-s − 3·19-s + 3·21-s − 6·23-s − 27-s − 6·29-s + 31-s + 33-s − 2·37-s − 39-s + 6·41-s − 11·43-s + 8·47-s + 2·49-s − 51-s − 6·53-s + 3·57-s + 4·59-s − 15·61-s − 3·63-s − 5·67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.242·17-s − 0.688·19-s + 0.654·21-s − 1.25·23-s − 0.192·27-s − 1.11·29-s + 0.179·31-s + 0.174·33-s − 0.328·37-s − 0.160·39-s + 0.937·41-s − 1.67·43-s + 1.16·47-s + 2/7·49-s − 0.140·51-s − 0.824·53-s + 0.397·57-s + 0.520·59-s − 1.92·61-s − 0.377·63-s − 0.610·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.22209059756078, −13.03373519476348, −12.44659817422804, −12.04755046849719, −11.67786442529532, −11.03090279725867, −10.50886108875921, −10.30819562311645, −9.711616602590133, −9.274418347563279, −8.843006780556873, −8.177422241383521, −7.641702932880017, −7.290974207824936, −6.552980652386042, −6.175148908090059, −5.936215394143588, −5.279465179918700, −4.685603090754479, −4.122995755780182, −3.569507551205602, −3.156681878767565, −2.363144252403507, −1.830164598328970, −1.070490666443568, 0, 0,
1.070490666443568, 1.830164598328970, 2.363144252403507, 3.156681878767565, 3.569507551205602, 4.122995755780182, 4.685603090754479, 5.279465179918700, 5.936215394143588, 6.175148908090059, 6.552980652386042, 7.290974207824936, 7.641702932880017, 8.177422241383521, 8.843006780556873, 9.274418347563279, 9.711616602590133, 10.30819562311645, 10.50886108875921, 11.03090279725867, 11.67786442529532, 12.04755046849719, 12.44659817422804, 13.03373519476348, 13.22209059756078
