| L(s) = 1 | + 2-s − 4-s − 2·7-s − 3·8-s − 4·13-s − 2·14-s − 16-s + 2·17-s − 4·19-s + 23-s − 4·26-s + 2·28-s − 10·29-s + 4·31-s + 5·32-s + 2·34-s − 2·37-s − 4·38-s + 10·41-s − 2·43-s + 46-s + 8·47-s − 3·49-s + 4·52-s + 10·53-s + 6·56-s − 10·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.755·7-s − 1.06·8-s − 1.10·13-s − 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.208·23-s − 0.784·26-s + 0.377·28-s − 1.85·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s − 0.328·37-s − 0.648·38-s + 1.56·41-s − 0.304·43-s + 0.147·46-s + 1.16·47-s − 3/7·49-s + 0.554·52-s + 1.37·53-s + 0.801·56-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286753204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286753204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−8.158419343959537475661275997633, −7.43918334459967781535831836234, −6.63550884652641025646177792791, −5.88397044187780158068176030942, −5.30541293154793434947141010050, −4.46527427055793605019210015089, −3.82617618923626444482449099861, −3.01788892315009954880909556281, −2.18373965623845132136210761823, −0.52823484155876893475860386148,
0.52823484155876893475860386148, 2.18373965623845132136210761823, 3.01788892315009954880909556281, 3.82617618923626444482449099861, 4.46527427055793605019210015089, 5.30541293154793434947141010050, 5.88397044187780158068176030942, 6.63550884652641025646177792791, 7.43918334459967781535831836234, 8.158419343959537475661275997633
