| L(s) = 1 | − 0.561·5-s − 5.12·7-s − 11-s − 3.12·13-s − 2·17-s + 4·19-s − 6.56·23-s − 4.68·25-s + 3.12·29-s − 1.43·31-s + 2.87·35-s + 3.43·37-s − 7.12·41-s − 1.12·43-s − 8·47-s + 19.2·49-s − 4.24·53-s + 0.561·55-s − 12.8·59-s + 7.12·61-s + 1.75·65-s − 5.43·67-s − 3.68·71-s − 3.12·73-s + 5.12·77-s − 2.87·79-s + 9.12·83-s + ⋯ |
| L(s) = 1 | − 0.251·5-s − 1.93·7-s − 0.301·11-s − 0.866·13-s − 0.485·17-s + 0.917·19-s − 1.36·23-s − 0.936·25-s + 0.579·29-s − 0.258·31-s + 0.486·35-s + 0.565·37-s − 1.11·41-s − 0.171·43-s − 1.16·47-s + 2.74·49-s − 0.583·53-s + 0.0757·55-s − 1.66·59-s + 0.912·61-s + 0.217·65-s − 0.664·67-s − 0.437·71-s − 0.365·73-s + 0.583·77-s − 0.323·79-s + 1.00·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5159338435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5159338435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 - 9.68T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.84960030077535263021306549704, −7.38182180226491932793983993091, −6.49375406684251816548198980593, −6.12615412230479030103308315268, −5.22049160386946861150690571969, −4.32733388686746680024234025036, −3.46975669874966716870517664579, −2.93720590940063199864601979211, −1.97344929397874887580778724500, −0.34799344596824056453445742882,
0.34799344596824056453445742882, 1.97344929397874887580778724500, 2.93720590940063199864601979211, 3.46975669874966716870517664579, 4.32733388686746680024234025036, 5.22049160386946861150690571969, 6.12615412230479030103308315268, 6.49375406684251816548198980593, 7.38182180226491932793983993091, 7.84960030077535263021306549704
