| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 2·13-s + 2·14-s + 16-s + 2·19-s + 23-s − 5·25-s + 2·26-s + 2·28-s + 6·29-s − 4·31-s + 32-s − 10·37-s + 2·38-s + 6·41-s + 2·43-s + 46-s − 3·49-s − 5·50-s + 2·52-s − 12·53-s + 2·56-s + 6·58-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.458·19-s + 0.208·23-s − 25-s + 0.392·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.64·37-s + 0.324·38-s + 0.937·41-s + 0.304·43-s + 0.147·46-s − 3/7·49-s − 0.707·50-s + 0.277·52-s − 1.64·53-s + 0.267·56-s + 0.787·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.251042667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.251042667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.28739002156340754631935353746, −10.61012688599013645156709530680, −9.437023229880634793301707393383, −8.316385130284391740534536615268, −7.47031472923997154911967627467, −6.34330275215127010693551770470, −5.35228133033607928968047127956, −4.38803397911697295812780705157, −3.20432850350217206557641382010, −1.67007626242106081786857040508,
1.67007626242106081786857040508, 3.20432850350217206557641382010, 4.38803397911697295812780705157, 5.35228133033607928968047127956, 6.34330275215127010693551770470, 7.47031472923997154911967627467, 8.316385130284391740534536615268, 9.437023229880634793301707393383, 10.61012688599013645156709530680, 11.28739002156340754631935353746
