| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s + 2·13-s + 15-s − 2·17-s + 2·21-s + 2·23-s + 25-s + 27-s − 4·29-s − 4·31-s + 2·35-s + 2·37-s + 2·39-s − 4·41-s + 10·43-s + 45-s − 6·47-s − 3·49-s − 2·51-s + 6·53-s − 4·59-s + 10·61-s + 2·63-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.718·31-s + 0.338·35-s + 0.328·37-s + 0.320·39-s − 0.624·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.520·59-s + 1.28·61-s + 0.251·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.262687276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.262687276\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.78686631382210, −12.06959609481217, −11.59138183099690, −11.12201500933565, −10.81735536019232, −10.32191461951231, −9.711593790618336, −9.366867378052447, −8.804326244053389, −8.577332357065795, −7.970743932668336, −7.578729737208758, −7.001196489983385, −6.658085239637484, −5.866277816949626, −5.648400621683150, −4.944318187002938, −4.530891990656272, −3.949736983461068, −3.477635482919383, −2.832442023322350, −2.288721834792666, −1.733092344860766, −1.313201528086230, −0.5044059495458763,
0.5044059495458763, 1.313201528086230, 1.733092344860766, 2.288721834792666, 2.832442023322350, 3.477635482919383, 3.949736983461068, 4.530891990656272, 4.944318187002938, 5.648400621683150, 5.866277816949626, 6.658085239637484, 7.001196489983385, 7.578729737208758, 7.970743932668336, 8.577332357065795, 8.804326244053389, 9.366867378052447, 9.711593790618336, 10.32191461951231, 10.81735536019232, 11.12201500933565, 11.59138183099690, 12.06959609481217, 12.78686631382210
