| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s + 13-s − 14-s + 16-s − 7·19-s − 20-s − 22-s + 3·23-s + 25-s + 26-s − 28-s + 8·29-s + 5·31-s + 32-s + 35-s − 5·37-s − 7·38-s − 40-s + 10·41-s + 2·43-s − 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.60·19-s − 0.223·20-s − 0.213·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.48·29-s + 0.898·31-s + 0.176·32-s + 0.169·35-s − 0.821·37-s − 1.13·38-s − 0.158·40-s + 1.56·41-s + 0.304·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.612055043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.612055043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.65004085631761, −12.36372285057628, −11.98810722148669, −11.33156121533634, −10.97870600801757, −10.48523859082791, −10.19685504956087, −9.549945715558996, −8.934411674240956, −8.420980376057115, −8.202570923014024, −7.489682451908651, −6.955773625273206, −6.634327138754811, −6.036030025773723, −5.755711151680829, −4.841258638945657, −4.661185116529271, −4.089224546070894, −3.573217681703387, −2.965988619330785, −2.527749385295097, −1.964081609912694, −1.065172539445649, −0.4912865537168265,
0.4912865537168265, 1.065172539445649, 1.964081609912694, 2.527749385295097, 2.965988619330785, 3.573217681703387, 4.089224546070894, 4.661185116529271, 4.841258638945657, 5.755711151680829, 6.036030025773723, 6.634327138754811, 6.955773625273206, 7.489682451908651, 8.202570923014024, 8.420980376057115, 8.934411674240956, 9.549945715558996, 10.19685504956087, 10.48523859082791, 10.97870600801757, 11.33156121533634, 11.98810722148669, 12.36372285057628, 12.65004085631761
