| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 2·11-s − 7·13-s − 15-s − 3·17-s − 6·19-s − 21-s + 3·23-s − 4·25-s − 27-s − 2·29-s − 3·31-s − 2·33-s + 35-s + 4·37-s + 7·39-s − 2·43-s + 45-s − 8·47-s − 6·49-s + 3·51-s + 2·53-s + 2·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.94·13-s − 0.258·15-s − 0.727·17-s − 1.37·19-s − 0.218·21-s + 0.625·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s − 0.538·31-s − 0.348·33-s + 0.169·35-s + 0.657·37-s + 1.12·39-s − 0.304·43-s + 0.149·45-s − 1.16·47-s − 6/7·49-s + 0.420·51-s + 0.274·53-s + 0.269·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9878605103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9878605103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 113 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.31748239855020, −15.04415380496792, −14.52788818572056, −14.07765732889222, −13.09214179170539, −12.96447423842959, −12.27930137745901, −11.62332789529162, −11.30788813137033, −10.58841079932257, −10.04964825072213, −9.473494610644504, −9.061139073902855, −8.192526594615736, −7.656576997835434, −6.791493075275344, −6.649796107314600, −5.735737188918530, −5.183312149411915, −4.530970377699946, −4.107685505191431, −2.998319416281691, −2.127607190801016, −1.720786045112125, −0.3984153853462602,
0.3984153853462602, 1.720786045112125, 2.127607190801016, 2.998319416281691, 4.107685505191431, 4.530970377699946, 5.183312149411915, 5.735737188918530, 6.649796107314600, 6.791493075275344, 7.656576997835434, 8.192526594615736, 9.061139073902855, 9.473494610644504, 10.04964825072213, 10.58841079932257, 11.30788813137033, 11.62332789529162, 12.27930137745901, 12.96447423842959, 13.09214179170539, 14.07765732889222, 14.52788818572056, 15.04415380496792, 15.31748239855020
