| L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 11-s + 4·13-s − 4·15-s − 2·17-s − 19-s + 4·23-s − 25-s + 4·27-s + 8·29-s − 10·31-s + 2·33-s − 6·37-s − 8·39-s + 12·41-s + 4·43-s + 2·45-s + 12·47-s − 7·49-s + 4·51-s − 14·53-s − 2·55-s + 2·57-s + 6·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 1.03·15-s − 0.485·17-s − 0.229·19-s + 0.834·23-s − 1/5·25-s + 0.769·27-s + 1.48·29-s − 1.79·31-s + 0.348·33-s − 0.986·37-s − 1.28·39-s + 1.87·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s − 49-s + 0.560·51-s − 1.92·53-s − 0.269·55-s + 0.264·57-s + 0.781·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.362150615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362150615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.832344002161694293680925777484, −7.82026973427424011527254022799, −6.84152193686306517428112570156, −6.22338084887670893500629726100, −5.69036806485513023902243370455, −5.04767866236744511097519285005, −4.12630917161403719032976863591, −2.96799215472902783423002395124, −1.87001429609774045393814183640, −0.74757916196460966029456081720,
0.74757916196460966029456081720, 1.87001429609774045393814183640, 2.96799215472902783423002395124, 4.12630917161403719032976863591, 5.04767866236744511097519285005, 5.69036806485513023902243370455, 6.22338084887670893500629726100, 6.84152193686306517428112570156, 7.82026973427424011527254022799, 8.832344002161694293680925777484
