| L(s) = 1 | − 3-s + 4·5-s − 7-s − 2·9-s + 5·11-s − 4·15-s − 2·19-s + 21-s + 23-s + 11·25-s + 5·27-s − 9·31-s − 5·33-s − 4·35-s − 11·37-s + 5·41-s + 8·43-s − 8·45-s − 9·47-s + 49-s − 8·53-s + 20·55-s + 2·57-s + 6·59-s + 5·61-s + 2·63-s + 13·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 1.03·15-s − 0.458·19-s + 0.218·21-s + 0.208·23-s + 11/5·25-s + 0.962·27-s − 1.61·31-s − 0.870·33-s − 0.676·35-s − 1.80·37-s + 0.780·41-s + 1.21·43-s − 1.19·45-s − 1.31·47-s + 1/7·49-s − 1.09·53-s + 2.69·55-s + 0.264·57-s + 0.781·59-s + 0.640·61-s + 0.251·63-s + 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.771678489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.771678489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.07066833006261, −13.70684713574621, −12.93249738411042, −12.61160105903754, −12.21334013975795, −11.43927450631219, −10.95896919429509, −10.69537199431057, −9.904246441380337, −9.495799068490032, −9.137667110555466, −8.707950813499924, −8.033810305449739, −7.011275398969111, −6.619895117485859, −6.363744660688911, −5.706470248770216, −5.326630876232317, −4.845627325856128, −3.836091927669149, −3.432502865749076, −2.505957209955762, −2.001744442983937, −1.359099766793608, −0.5745507075014465,
0.5745507075014465, 1.359099766793608, 2.001744442983937, 2.505957209955762, 3.432502865749076, 3.836091927669149, 4.845627325856128, 5.326630876232317, 5.706470248770216, 6.363744660688911, 6.619895117485859, 7.011275398969111, 8.033810305449739, 8.707950813499924, 9.137667110555466, 9.495799068490032, 9.904246441380337, 10.69537199431057, 10.95896919429509, 11.43927450631219, 12.21334013975795, 12.61160105903754, 12.93249738411042, 13.70684713574621, 14.07066833006261
