| L(s) = 1 | + 3-s + 4·7-s + 9-s + 6·13-s + 2·17-s − 6·19-s + 4·21-s + 6·23-s + 27-s + 8·29-s − 8·31-s + 10·37-s + 6·39-s − 6·41-s + 4·43-s + 2·47-s + 9·49-s + 2·51-s − 6·53-s − 6·57-s + 12·59-s − 14·61-s + 4·63-s + 4·67-s + 6·69-s − 8·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.66·13-s + 0.485·17-s − 1.37·19-s + 0.872·21-s + 1.25·23-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 1.64·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.291·47-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 0.794·57-s + 1.56·59-s − 1.79·61-s + 0.503·63-s + 0.488·67-s + 0.722·69-s − 0.949·71-s + 0.468·73-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
| \(L(1)\) |
\(\approx\) |
\(3.902503811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.902503811\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.914685111449344206213296030654, −7.09837284674024644967711361550, −6.34178716138858146762520038395, −5.63286561542429334939347390651, −4.77171961481171123571145341753, −4.24424039728167433212118356314, −3.46279301870496658414399762005, −2.56376769526120324546512750842, −1.64374345802503486358222192097, −1.02430987920486690742915538441,
1.02430987920486690742915538441, 1.64374345802503486358222192097, 2.56376769526120324546512750842, 3.46279301870496658414399762005, 4.24424039728167433212118356314, 4.77171961481171123571145341753, 5.63286561542429334939347390651, 6.34178716138858146762520038395, 7.09837284674024644967711361550, 7.914685111449344206213296030654
