| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s + 2·13-s − 15-s + 6·17-s + 8·23-s + 25-s − 27-s + 10·29-s − 8·31-s − 4·33-s + 2·37-s − 2·39-s + 2·41-s − 8·43-s + 45-s + 4·47-s − 6·51-s + 10·53-s + 4·55-s + 4·59-s + 6·61-s + 2·65-s − 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 1.21·43-s + 0.149·45-s + 0.583·47-s − 0.840·51-s + 1.37·53-s + 0.539·55-s + 0.520·59-s + 0.768·61-s + 0.248·65-s − 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.709704431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709704431\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.59041479133974, −16.00761736812101, −15.18637729794203, −14.69290283787892, −14.16652283270193, −13.58470285563423, −12.91648060187652, −12.32639450517970, −11.87955828801392, −11.18031982573262, −10.72196998481654, −9.973067258496156, −9.515997426681493, −8.808437501946534, −8.273662808322279, −7.297483823779897, −6.770149661124197, −6.258572922553006, −5.425865707345083, −5.071996247237201, −4.029016854511189, −3.468179712827011, −2.540057923681851, −1.328368080170057, −0.9316105087720988,
0.9316105087720988, 1.328368080170057, 2.540057923681851, 3.468179712827011, 4.029016854511189, 5.071996247237201, 5.425865707345083, 6.258572922553006, 6.770149661124197, 7.297483823779897, 8.273662808322279, 8.808437501946534, 9.515997426681493, 9.973067258496156, 10.72196998481654, 11.18031982573262, 11.87955828801392, 12.32639450517970, 12.91648060187652, 13.58470285563423, 14.16652283270193, 14.69290283787892, 15.18637729794203, 16.00761736812101, 16.59041479133974
