| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 7·13-s + 15-s − 6·17-s + 3·19-s − 2·23-s + 25-s − 27-s − 2·29-s + 7·31-s + 4·33-s − 7·37-s − 7·39-s − 8·41-s + 5·43-s − 45-s + 10·47-s + 6·51-s − 8·53-s + 4·55-s − 3·57-s + 10·59-s − 6·61-s − 7·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.94·13-s + 0.258·15-s − 1.45·17-s + 0.688·19-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.25·31-s + 0.696·33-s − 1.15·37-s − 1.12·39-s − 1.24·41-s + 0.762·43-s − 0.149·45-s + 1.45·47-s + 0.840·51-s − 1.09·53-s + 0.539·55-s − 0.397·57-s + 1.30·59-s − 0.768·61-s − 0.868·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152936244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152936244\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.558946721771150800568741434232, −8.134495411419646647294711819114, −7.16739230183214880415590090208, −6.44487685730593374651625866722, −5.71692474374451453974731774699, −4.91097050490698175564711466411, −4.05741569009759713093068762395, −3.21891914623107077829001634948, −1.99894069041554242253603743250, −0.67222157365776090445570599748,
0.67222157365776090445570599748, 1.99894069041554242253603743250, 3.21891914623107077829001634948, 4.05741569009759713093068762395, 4.91097050490698175564711466411, 5.71692474374451453974731774699, 6.44487685730593374651625866722, 7.16739230183214880415590090208, 8.134495411419646647294711819114, 8.558946721771150800568741434232
