| L(s) = 1 | + 0.445·2-s − 1.24·3-s − 0.801·4-s − 0.554·6-s − 0.801·8-s + 0.554·9-s + 12-s + 0.445·16-s + 0.246·18-s − 0.445·19-s + 24-s + 0.554·27-s + 1.24·29-s + 32-s − 0.445·36-s + 0.445·37-s − 0.198·38-s + 1.80·43-s − 0.554·48-s + 49-s + 0.246·54-s + 0.554·57-s + 0.554·58-s + 71-s − 0.445·72-s + 1.80·73-s + 0.198·74-s + ⋯ |
| L(s) = 1 | + 0.445·2-s − 1.24·3-s − 0.801·4-s − 0.554·6-s − 0.801·8-s + 0.554·9-s + 12-s + 0.445·16-s + 0.246·18-s − 0.445·19-s + 24-s + 0.554·27-s + 1.24·29-s + 32-s − 0.445·36-s + 0.445·37-s − 0.198·38-s + 1.80·43-s − 0.554·48-s + 49-s + 0.246·54-s + 0.554·57-s + 0.554·58-s + 71-s − 0.445·72-s + 1.80·73-s + 0.198·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6624548197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6624548197\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 0.445T + T^{2} \) |
| 3 | \( 1 + 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.80T + T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 - 0.445T + T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.533809117492334232905616423481, −8.749676887549675169869034214845, −7.933467742952561953343259587929, −6.78231852590847581561049931265, −6.06543360272651378350888864909, −5.41491449903659888756647206503, −4.66445162884313331504616397613, −3.94666446874633105705542335049, −2.67976955216007079673286458685, −0.827836044704786324084222436229,
0.827836044704786324084222436229, 2.67976955216007079673286458685, 3.94666446874633105705542335049, 4.66445162884313331504616397613, 5.41491449903659888756647206503, 6.06543360272651378350888864909, 6.78231852590847581561049931265, 7.933467742952561953343259587929, 8.749676887549675169869034214845, 9.533809117492334232905616423481
